If you can gather up a few of the common failure parts for CRTs (especially if you know the weaknesses of your own CRT), you should be able to keep it running for a good long while (vertical deflection IC, horizontal output transistor, certain zeners, a couple cap kits, etc). Of course, some failures may put the cost beyond reasonable economic repair, but that's true for LCDs too. With the right parts you can usually keep a halfway decent CRT running for a decades. I know people with CRTs from the 70s and 80s that still work.
CRT's offer a more authentic arcade experience, as well as having softer edges, a kind of inherent anti-aliasing. This is why, in my opinion, CRTs are usually the better choice for a home arcade machine.
However, LCD's do offer a smaller profile, lighter weight, easier connectivity, and great color reproduction which makes them a viable option if you don't mind the added pixelation.
In other words, I would suggest repair over replacement.