MacBook Pro 13" Unibody Mid 2010, can the video card be upgraded?

Not easily it is all part of the main logic board. The question is why? Card integrated was designed & developed for the board. If you have maxed out your RAM. You could put in a 2nd quicker hard drive. Installing MacBook Pro 13" Unibody Mid 2010 Dual Hard Drive

No computer lasts forever, however well you maintain it or look after it. My advice would be to put any spare cash put towards a newer machine.

To squeeze any extra performance out of the machine the best thing you can do it to use a de-installer to remove any software you don't use on a regular basis. (I use Cleanmymac, but there are others out there.)

Plus don't have millions of files on the desktop use actual aliases. (CMD-L)

Sorry the GPU logic is part of the Haswell CPU and not upgradable in the 13" models.

The 15" MacBook Pro models have a dedicated GPU which offers better graphics performance.

To gain the best graphics performance you'll need to upgrade your system to a Thunderbolt2 or 3 based system and then use an external chassis to plug in a high end graphics board.