Once at your download list, click “Download to your PS Vita” on the game that you want.
If you haven’t bought the game that you want, do that.
SCE has not gone on record about the update schedule for PSP, but a few major PSP features are still MIA -- most importantly, the aforementioned Play Station Store downloading feature as well as the assumed ability to connect to the Play Station Network community that PS3 currently uses.
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A new version of the PSP firmware, version 3.50, is now available, updating PSP's Remote Play capabilities.
The firmware update occurs automatically and it is the easiest way to upgrade the PSP.
To update PSP firmware for the PC, the PSP has to be connected to the PC.