The Romans themselves thought that the elected magistracies originally were only open to patricians, although this tradition is probably inaccurate. In point of fact even the consulship was almost certainly open to plebeians from day one--of the original consuls Brutus was a plebeian, from the plebeian gens Junia. But in any case by the end of the Conflict of the Orders, and in particular after the passage of the Lex Licinia in 367, which made it illegal for one of the two consuls to not be a plebeian, the Roman nobility had certainly shifted from its old form. By 300 plebeians could even hold pontifical office, with the exception of a small handful of priesthoods that probably dated back to the kingship. Part of the problem I think is that you're not defining what you mean by the "upper classes." Do you mean the patrician order, which had essentially died out by the 1st Century (with only 14 remaining families by Caesar's lifetime) and had ceased to have any important political privileges? Or do you mean the senatorial class, composed of both patricians and plebeians? In either case the answer is no, elected magistracies were technically open to any Roman citizen possessing the necessary amount of property (which, although a chunk of change, was hardly a prohibitively high amount). But beyond that simple response the realities of the situation change depending on what exactly you're asking. There was essentially no difference between the patricians and senatorial plebeians after the resolution of the Conflict of the Orders, but if we're talking about the senatorial class as the "upper classes" (which is how most scholars of the Roman Republic would mean, although increasingly I find that elsewhere people usually mean the patricians for some reason, hence my confusion here) then the situation is much more complicated. Though there was no legal prohibition against someone of enough means running for office it was not something that happened much--there are only a handful of novi homines, those men elected to office without a history of senatorial magistracy, from the end of the Punic Wars to Cicero, himself a novus homo. It's not that there was any legal issues with a wealthy plebeian (recall that there's also a problem with describing plebeians. The plebs are not necessarily the same thing as the plebeian order, and the plebeians are not really a social class--as the New Pauly says, "Only the relatively homogenous patriciate should here be understood as an 'order'." And of course they had died off--by Caesar's day just about everyone was a plebeian, thought not necessarily one of the plebs), it's just that canvassing for office was hard and if you didn't have support you weren't going to win.

Elected magistrates (there were a few minor senatorially-appointed magistrates, but they're rarely important other than the dictatorship, an office that ceased to exist after the Punic Wars--Sulla's "dictatorship" and Caesar's were both really extraordinary magistracies carrying the same name) were not nominated by anybody, but declared their own candidacy of their own volition. Without getting too much into it candidates could begin campaigning for office years before their intended electoral year--Cicero began his canvass for the consular elections for 63 in 65--and were expected to gather support throughout all levels of society. After all, magistrates were elected by the votes of the comitia, so therefore it was necessary to ensure support throughout all levels of Roman society. A candidate for office had to possess a certain amount of property (the precise amount changed over time, although it was never especially crippling) and though that was pretty much the only legal requirement he wouldn't get too far if he wasn't a good orator and didn't have the reputation and connections fitting for his intended office. The only senatorial magistracies which were not open to the plebeians were the curule aedileships, which many politicians skipped entirely and which had a perfectly good (and possibly more important, since they did more than the curule aediles) counterpart in the plebeian aedileships.

But again, this is sort of an unclear question because it's not obvious what you mean by the "upper" and "lower" classes. You speak of the "plebs," but that's not a term that means very much, it's like saying the citizen body as opposed to Congress. An equestrian like Cicero, that is to say those non-senatorial plebeians possessing the property and education to run for elected office but whose families never had and did not themselves, might be elected as a novus homo but is that what you mean by "plebs?" If you mean the plebeian order, that is those not descended from the patres, they made up the majority of the senate and thanks to the Lex Licinia at least half the nobiles, those with consular ancestors. It's really rather hard to find a Roman politician of Caesar's time who's not a plebeian--Hortensius, Brutus, Cassius, Antony, Cicero, Crassus, Cato, Lepidus, Lucullus, Pompey etc. are all plebeian names. But not just any random dude off the street could run, because of the property requirement--that's why the equestrian order, originally a sort of unofficial label but under Augustus an order legally separate from the plebeians and patricians, existed at all.

Oh and the other position which the same question applies to is Censor.