Can I manually update my display driver? I've recently acquired a MacBook Pro. It's a 13 inch model which comes with an Intel HD Graphics 3000 display card.

I've had a few issues lately (such as flickering graphics in certain games and on youtube when hardware acceleration is used), and what I would do on a Windows machine is simply download and install the latest drivers to see if that helps.

Is this even an option when using a Mac? Or is this something I can't (and don't need to) worry about?
There are no driver updates for components found within a MacBook Pro (or any Mac for that matter) found anywhere but through Apple's own update system. And as you say, it's not something you need to worry about.

They retain tight control over their components and mainly make driver changes through OS updates rather than parse everything into individual packages. However, they may release small updates that attend to these kinds of glitches periodically or when a serious issue arises.

So long as Software Update reports no new updates, you will be running the latest drivers available for your Mac. You won't find alternate drivers elsewhere.

Additionally, what you describe is not normal and I would suggest you see a Genius at an Apple Store (not a licensed vendor). They will investigate and you may just walk out with a new machine. Apple has legendary service; make use of it.