I got banned on mineplex. If I buy a new minecraft user can I use it again on the same pc?

Depends on the type of ban they utilized…

If it was just a simple username ban, buying a new premium account would allow you to bypass said ban.

If it was an IP ban (which is generally reserved for more serious offenses), you're out of luck as their banning your connection directly.

Depending on the offense, it's possible they may have done both.

I Can't RECOMMEND ATTEMPTING TO BYPASS ANY BANS IMPOSED UPON YOU WITHOUT GOING THROUGH THE NECESSARY APPEALS PROCESS BY THAT SERVER (IF THERE's A PROCESS AVAILABLE)!

Bypassing a ban is as serious (& possibly WORSE) than the original offense that cause them to issue the original ban. You would be held in contempt of the server's administration & they would have every right to impose even harsher bans on your new account, would easily include username ban as well as an IP ban. If they were really ticked off, they might even go as far as issuing an IP range ban (which can cover your entire ISP) to ensure you can't get back on their servers.

While spending $27 is bad enough to your wallet, IP range bans can cost you hundreds (to change ISP's) to potentially thousands (to physically move to a new town w/ a different ISP). I know there's more cost effective options to bypass IP bans, but you're dealing additional subscription fees & whatnot that's generally more hassle on your side than their server admin's side.

The server admins would have to consider the potential fallout with IP range bans as well, since they can affect innocent users as well… But can be deemed a necessary evil when dealing people who refuse to honor their decisions & go away.

Please note that 1st amendment rights only protect you from the government from censoring your right to free speech. It doesn't extend to private domains (like game servers) & them issuing a ban is their way of showing you the exit as they don't want to deal with you anymore… So TAKE THE HINT AND LEAVE THEM ALONE!