A good soldering iron is nice to have, one that heats up quickly and outputs enough heat to properly solder components on large ground planes.
A good multimeter can perform most basic trouble shooting on an arcade monitor. Check the B+ voltage, the heater voltage, diodes, approximate resistance values at certain areas, quick check if caps or transistors are shorted, etc etc. Must have item right there.
A desoldering gun, vacuum powered preferably, makes a *huge* difference for removing parts. I have a Hakko FR301, and it makes repairs as well as testing (some components need to be removed or have one leg disconnected from circuit before testing) so much faster. Probably the single largest quality of life improvement I've bought.
A test pattern generator is extremely valuable, but there are a couple mistakes one can make here. For example, don't consider it 'saving money' to buy a 20-30 dollar vintage TPG on ebay, because with shipping more than likely you will spend around 50 bucks, and when you do get it, it will likely be old and in need of repair/restoration, whether sticking buttons or old caps. After you purchase those items, pay shipping, and do all the work to get it working nice, you could have just bought a craftymech tpg for 90, and its the size of a small multimeter, easily put away in a box of test gear, while those vintage TPG's likely take up half a table.
An oscilloscope is pretty valuable when getting deeper into repairs. Most manuals will have a picture list of o-scope screens showing what the signal should look like at various parts, which helps narrow down the area with the difficulty a lot faster.
A good CRT Tester can be hard to find, they are certainly useful, but a good one is going to cost you a pretty penny. There are certain brands and models that are recommended over others. For example, most people say the B&K 467 and above are good choices. They are useful when trying to check the quality of a tube, how much life is left, what level of emissions it puts out, if there are shorts, etc. But, rejuvenation is a mixed bag. Its one of those things where some people say if you need to rejuv the tube, it probably should be replaced, as even if you're successful, there's no guarantee you will get much life after. The response is that if you're thinking to replace it, then risking a rejuv is acceptable, and its possible you could get years of use after. Some say that if you don't have a compatible tube to replace it with, don't rejuv. But the risk is up to you.
I'd recommend an ESR and Ring tester, but someone already mentioned them.