Ok so there's a lot going on here and you've got some of the events as described in Acts wrong. In Acts Paul does several things with his Roman citizenship. First, he protests against the decision to have him flogged on the grounds that it is unlawful to flog a Roman citizen who has not been condemned in a Roman court. Paul does not at this stage make any sort of official appeal or invoke any official power, he merely tacitly threatens that he will if the soldiers do as they're threatening. Paul is then taken to the Sanhedrin in order to figure out why exactly folks in Jerusalem want him dead, after which he is returned to the barracks. When a conspiracy is revealed against Paul he is transferred with some difficulty to Caesarea to answer to the procurator Felix. Felix is informed that the reason Paul is here is that there's a conspiracy against him and they had to get him out of Jerusalem, but also because the reason why he was arrested in the first place was that the people in Jerusalem wanted to kill him, apparently because of a matter of Jewish law. Since the provincial officials on site cannot determine whether, by Jewish law, he's supposed to be killed or what, they've decided that the case should be presented before the procurator to figure out what to do. Felix hears the case, but makes no judgement on it: instead, he keeps Paul imprisoned for two years until he is replaced by the new procurator Festus. Festus starts the process over again, and says he will send Paul back to Jerusalem to be tried. Paul insists that he has not broken Jewish law and, apparently worried that the court will not give him a fair trial, now appeals to Caesar. Now King Agrippa appears and, hearing about the case from Festus, asks to hear Paul's case. Festus explains to Agrippa that he can't find any good reason that Paul should be executed (it's unclear to me whether he's talking about Roman or Jewish law, but likely by the latter, especially since Paul's defense is that he hasn't broken Jewish law in any way) and that he would like Agrippa to help him figure out what exactly the Jewish case against Paul is so that he can inform the emperor. Agrippa listens to Paul's defense and says to Festus that had he not appealed to Caesar he could have been freed.

First things first. Paul does not at any point appeal to Agrippa, and it does not seem that he has the ability to do so. Agrippa is the king of Judaea, he's not a Roman official, and he has no power to do anything with Paul once Paul has invoked his rights as a Roman citizen. Paul is also not on trial when he speaks to Agrippa. The king is merely listening to his case in order to explain to the governor how Paul would have been treated in a legitimate court by Jewish law. The conclusion Agrippa comes to is that Paul has done nothing seriously wrong, whether by Jewish law or Roman law it's not entirely clear, and that in a legitimate court he would have been acquitted. However, there's a problem. While Agrippa judges Paul to be undeserving of death, he no longer has the power to make that decision. Paul has appealed to Caesar, and therefore is now the problem of the Roman courts. And it's unclear whether Agrippa would have had the power to judge Paul in any case. When Festus proposes to send Paul to stand trial in Jerusalem, he mentions that he himself will be there (the parallel with Christ is obvious in the passage). So while Agrippa's interpretation of the case is that Paul should be acquitted, his decision doesn't really matter, and Paul clearly fears an unfair trial if he is tried at Jerusalem.

In sum, there are two things at play here. The first is the clash between Roman law and local law. Roman citizens were subject to Roman law, ius civile, and to ius gentium, the "law of all peoples," a sort of universal concept of general legal practices that all peoples broadly were thought of as following (e.g. slavery happens). However, a Roman citizen living outside of Italy also had to deal with non-Roman local law, which was typically left intact when the Romans took over an area, though Roman officials were frequently inserted into the process. In very broad strokes, whether a Roman citizen accused of a crime abroad was liable to local law or Roman law was a very fraught question going back into the Republic, and while a complex system arose in the late Republic and Principate as to how to deal with these tensions, it was not entirely systematic and in the first century AD would have been particularly confusing. It's not entirely clear what Festus was proposing here. Either he's proposing that Paul be tried by a Jewish court, in which case Paul asserts his rights as a Roman citizen to a Roman court, or he's proposing that Paul be tried by Roman court, but in Jerusalem, where the court will be under pressure by the Jews. The end result is basically the same: Paul decides that local justice will treat him unfairly, and appeals to the emperor, in which case he must be taken to Rome.

The second thing is the appeal itself. Appeals to the emperor are a bit of a shady business, falling under a quasi-official power of the emperor that ultimately derived from (and shared terminology with) the old Republican powers of appellatio and provocatio. The latter is well understood and clear to us. Provocatio was in origin a sort of self-help, in which a citizen, under compulsion from a magistrate, called out (provocare) to the people. In its earliest forms the people quite literally came to the citizen's aid, in our sources sometimes violently. In later times the call to the people was often replaced by a formula in which the citizen appealed (appellare) to the tribunes (or sometimes to the tribunes and people together), and no matter what the specific type of formula the appeal, if accepted by the tribunes (it was not always), would be brought before the assembly for a vote, which might overturn the magistrate's coercion. This is the basis for Paul's first use of his citizenship, to protest the decision to flog him. By various laws of the Republic citizens could not be beaten except under certain circumstances so long as they invoked provocatio (the right was not assumed, but had to be physically invoked, though Paul's threat alone appears to have been enough) and a vote of the people confirmed their appeal. The other power, appellatio, is more relevant. Provocatio required a vote of the people, as well as acceptance of the appeal by the tribunes (since they would be the ones to bring the matter before the assembly for a vote). In the Principate the ability of the tribunes to accept appeals was tacitly extremely limited, and the assemblies were no longer convened really at all, but certainly not to decide cases of provocatio. A similar power, appellatio, existed and did not require a vote of the people. Appellatio is extremely hazy for us, as very few examples of it exist in the Republic (as opposed to provocatio, which is extremely common). It looks like the ability to call on (appellare) a higher magistrate to counteract the decision of a lower one was possible in a shadowy procedure called appellatio, but on what grounds is not known to us. Two major cases of appellatio in the Republic exist for us. One was urban. In 77 BC a eunuch attempted to bring a case before the praetor, at which point the consul was called on (appellatus) to make the judgement about whether a eunuch could even bring such a case before the praetor--the consul decided he couldn't. The other was provincial, and somewhat closer to Paul's circumstances. In Sicily a man brought a case before Verres the governor, and called on Verres to move the case up to a court in Rome, out of fear that Verres, who was notoriously corrupt, would appoint a rigged jury. That seems very similar to what Paul is doing--he wants to avoid what he argues would be an unfair trial, either by a Jewish jury or in Jerusalem while Jews are present. In any case the appeal failed. For some reason, either legally or otherwise, Verres seems to have simply ignored the appeal.