"Get him arrested" is covering a broad range of outcomes. Most answers thus far are dealing with whether the provocateur has technically committed a crime, or whether the puncher has a legal defense likely to prevail at trial. Both are relevant, but elide aspects of the process that are essential to the question posed.
The goal as stated is causing an arrest. If the scene is staged in front of a police officer, that's highly likely to occur as the puncher will at least be prevented from leaving until the situation has de-escalated and the parties have been separated. As an arrest is simply "the use of legal authority to deprive a person of their freedom of movement", this satisfies the requirements. But there's no certainty of jail time for either parties involved.
Officers have broad discretion in handling most situations. An arrest isn't even guaranteed in the situation described, if the officer is able to end the altercation with a verbal warning and ascertains that no serious threat or harm exists. They may arrest either or both parties to the incident (in the sense of detaining on site for a matter of some minutes), then let them go. They may issue citations to one or both parties for some variation of disorderly conduct or breach of peace. And they may bring one or both parties to the station, formally book them, and subject them to detention in a holding cell until they make bond (varies on jurisdiction, but generally for minor offenses bail is based on a pre-established schedule and can be paid without requiring a judge or court appearance) and/or are deemed sober enough to pose no threat. This last is likely what the questioner means by "get arrested" and it is by no means guaranteed, especially if the officer witnessed the provocation.
It's worth noting that some states do have "mandatory arrest" laws, which remove an officer's discretion in certain assaults. But to my knowledge those are all specifically related to domestic violence calls, which would entail a rather more elaborate and prolonged level of preparation and provocation than the question considers.
The question also mentions the punched person's "ability to insist [the puncher] be arrested", which seems to reflect a common misconception. Despite what television programs have led many to believe, a victim has no ability to insist that anyone be arrested, under any circumstances.
Typically, if you are the victim of a crime, you or someone calls the police, who report to the scene and record information about what happened. Police reports are then turned over to the office of the local state or district attorney and a lawyer is assigned to the case. That prosecuting lawyer reviews the police reports, including victim and witness statements, to decide whether charges should be filed.
If an officer witnessed and intervened in the fight as discussed above, the report is already made. But in either case, it's up to the DA's office to decide whether to pursue the matter further. If the report or their initial investigation turns up obvious provocation, this will factor into their consideration of whether or not to pursue the matter and formally press charges. For the puncher to be arrested after the fact, the DA's office would need to decide to press charges, and those charges would have to be sufficiently serious to justify pre-trial detention instead of a simple summons or notice to appear before the court. The specifics vary widely based on location and circumstances, but arrest warrants are not commonly issued for low level misdemeanors.