Some definitions are in order here. First, patricians were people said to be descended directly through the male line from one of Romulus' original 100 advisors or one of the very few people who had been entered into the roll of patricians at a later date (e.g. the Claudians). In Caesar's time there were almost none of them left, and by the end of the 30s there were so few that Augustus elevated many plebeian nobles (such as the gens Sempronia) to patrician status to prevent the various religious rituals attached to patrician families from dying out. Everyone who wasn't descended directly from one of these nonexistent guys is a plebeian. But wealthy and important plebeians were not something that had only recently begun to happen "by the time of Caesar." Wealthy plebeians and the plebeian nobility had always existed at Rome. Traditionally the Conflict of the Orders of the most ancient period at Rome was in large part kicked off by the desire of the plebeian aristocracy to be afforded the same political privileges as the patricians, especially candidacy for the consulship. Whether that's actually true or not doesn't matter much, because even if it's not then probably plebeians were always eligible for the consulship--of the first consuls one, Brutus, was a plebeian.

Patrician status was not attached to wealth, property, or anything else other than family. While a censor might eject someone from the senate for being profligate with his finances--though not, as is often supposed, for being below a magical property requirement, as the property qualification for the senate is a feature of the post-Augustan period--the only way to lose patrician status was either to be punished by deminutio (which probably never happened. To my knowledge there aren't any really secure cases of patrician families being punished with plebeian status) or by seeking a transfer to the plebs, which we know happened but of which we have very little understanding at all. Therefore yes, it was entirely possible to be a poor patrician.

Also depending on what you mean by "grain dole" then yes, patricians would have gotten that. If you mean the state-subsidized grain instituted by C. Gracchus and made free to all citizens by P. Clodius then obviously patricians, being citizens, received the state-subsidized grain, although it was pretty small amount even for an ordinary laboring family. Pompey's cura annonae a year later reduced the number of recipients to 320,000, so probably not a really big change. We can probably expect that senators were no longer allowed to receive the state-subsidized grain, which likely eliminates most, if not all, patricians. The grain was reduced further by Caesar, who had the praetors draw up a waiting list. The state-subsidized grain fluctuated under Augustus, but importantly from Augustus on the possessors of the state-subsidized grain received a metal ticket as proof. These were supposed to be inalienable--which is to say that once on the list one could not be struck from it--but we find them passed down in wills or sold, meaning that access to state-subsidized grain became a privilege under the emperors, not a right. It doesn't seem unlikely that some of these ended up in the hands of patricians, and it's almost certain that many of them ended up in the hands of their direct dependents, which is in some ways basically the same thing.

But the idea of "poor" and "wealthy" nobles doesn't, frankly, make very much sense. The upper levels of Roman society, like the upper levels of most societies, was too complicated to reduce a person's wealth to a mere calculus of his property and cash. Roman society, like all ancient societies, was a debt society. In the absence of electronic banking debt was and had to be the primary means by which any kind of large purchase was made. Moreover, aristocratic society is in any culture extremely expensive. Senators and magistrates were expected to spend lavishly from their own pockets, and magistracy was an opportunity to forge ties by means of debt. Hence the idea that Caesar in his youth was somehow impoverished, which doesn't seem to be the case at all, because he had to take out large debts, and hence how Catiline, despite having enormous wealth, also had so much debt that he embarked on conspiracy for tabulae novae. People like Sulla, who is said by Plutarch to have grown up in an insula below a family of freedmen, may have been poor on paper, but in many cases could have called on a great deal of wealth through debt, obligation, and other means.

Oh, and by the way, Sulla doesn't seem to have been poor. He seems to have had what you might call "moderate means." Plutarch claims that Sulla (like personally, he doesn't seem to mean Sulla's family) paid HS 3000 for their lodgings whereas the freedman living above him paid HS 2000, both substantial sums of money. If we take Cicero's claim that an unskilled slave could still make 3 denarii per day in the forum (almost certainly too high a figure) then Sulla's rent alone was about 3/4 of a laborer's expected yearly income assuming that the laborer worked every single day of the year, which is both impossible and improbable. So the dude wasn't well off, but he wasn't poor. I grew up that poor, it's a world of fucking difference. From his step-mother and some rich old lady that Plutarch talks about he apparently inherited enough property to run for the quaestorship, so it doesn't exactly sound like the dude didn't have a pot to piss in.

Sulla's example, however, brings up the actual problem of status. A noble family's fortunes had little to do with plebeian or patrician status and everything to do with whether they had lapsed into obscurity. Such things were not totally uncommon, and especially among the plebeian nobility it sometimes happened that young men of senatorial families decided not to go into politics and opted instead to remain equites (e.g. the satirist Lucilius). Such things are much less certainly attested among patrician families, but that probably is due more to the fact that there were so few patrician families than anything else. Sulla's family was less fortunate because they had fallen from political prominence. It's unclear what that means exactly, and it comes mostly from Plutarch. We don't know whether anyone in Sulla's family after the ejection from the senate of his ancestor the consul Rufinus in 275 attained a magistracy or (more likely) was included in the senate. Hell, we don't know what happened to Rufinus after he got booted from the senate. He may well have returned before his death, possibly following the same route as Lentulus of Catilinarian fame by winning a magistracy like the praetorship, though that seems unlikely. It doesn't seem far-fetched to think that some of Sulla's ancestors must have reached at least the quaestorship or attained a senatorial position, which for much of the time between Rufinus and Sulla would not have been explicitly tied to a former tenure as a magistrate. It seems hard to believe that even though Sulla's family had lapsed from prominence a couple centuries ealier they didn't maintain some political presence, since it becomes hard to explain Sulla's easy climb through the quaestorship and military tribunate--for both of which he was given choice assignments, under Marius in Africa--if he was a genuine nobody. Further, things could get weird when a patrician man was young and his property technically belonged to his father, or before his mother died and he could inherit her property. There's a great deal of evidence to suggest that senatorial youths, before embarking on their political careers, not infrequently ended up taking on large amounts of debt, probably for all the reasons I've already mentioned.

So is the scenario you're talking about possible? Yes, although it's not terribly probable, and the examples usually given of "poor" patricians simply aren't convincing. Still, it's probably asking the wrong question. For a patrician family to be truly destitute it would have had to have lost all its connections, if for no other reason than that patricians were constantly inheriting each other's wealth. A family that had so significantly dropped out of the circle of the nobility would most likely die off pretty soon. And just because a family is not "prominent" doesn't mean that they're not connected. We don't hear anything at all, for example, of the patrician gens Sergia between Sergius Silus (praetor 197) and his great-grandson Catiline, but it's quite obvious that they remained a connected, if politically unimportant, patrician family. And sure, Sallust says that Catiline suffered from "inopia rei familiaris" (poverty of his family property), but what constituted inopia was very different for someone like Catiline, whose position required him to spend hundreds of thousands of sesterces annually, than for the common laborer on the forum building projects.