So I installed NPS Browser on my PC and set it up all right my question now is how do I get the game to show itself as a Bubble on the Vita homescreen
It's not a PSX Game its a VITA Game
After downloading, NPS Browser (in its default settings) should have extracted the game to a folder, named as the TITLE ID of the game. For example, Phantasy Star Nova is PCSG00351, so there should be a folder named like that.
Copy this folder via Vitashell to your ux0:app/ folder. If you have other games installed, you will notice that this "app" folder has various folders named with a similar structure (PCSG00351, PCSG00141, etc.). These are the games/apps installed on your Vita.
After that, you just need to go to the root of Vitashell (from where you can see all of the partitions, like ur0:, ux0:, etc.), press triangle, and select ""Refresh Livearea..."; wait for it to finish and it should say "Refreshed X items...", with X being the number of games (or DLC) that you just copied and Vitashell detected (here you can check if what you copied wasn't recognized when you read how many items were refreshed).
Afterwards, you can go to to your Livearea and the bubble should be there. I'd recommend rebooting your Vita anyway, as some games require a reboot for the DLC to be recognized, though this is extremely rare (you can reboot from Vitashell if you don't want to go back to your Livearea, too).
And, that's all. Your game is now on your Livearea. You can update it from there if you want.