If you are not planning to drill a new hole for a new tonearm, you can get away with:
1) the stupid protractor downloaded off vinylengine - print to scale, stick transparent tape, cut the spindle hole, and use for null point alignments. Extremely useful tool.
2) a 6 inch steel ruler accurate to half mm to measure and adjust cartridge overhang. In my limited experience, I find that the overhang doesn't have too much effect on sound quality. If you can get roughly about 15.3 mm or so, it works fine for most cartridges.
3) spirit bubble to measure if azimuth is right. This must be small and light enough to be mounted on top of headshell. Closing one eye and eyeing the headshell-cartridge combo from the front of the cartridge with the open eye, taking due consideration of parallax, also works quite well for checking azimuth. Anyway, this is moot if your arm doesn't allow azimuth adjustment.
4) a long spirit level to check if your table is level. This is often over looked but a critical foundation.
5) a set of allen keys to unscrew/tighten counterweight, antiskate, etc (you may not even need this depending on your arm model).
6) small flat as well as star screw drivers to loosen/tighten headshell screws. As far as possible, I try and restrict myself from using screw drivers for fixing cartridges. Finger tightening is usually sufficient in most cases.
7) a small brush to periodically brush off dust and hair-like things accumulated on the stylus. The brush that comes with the Denon DL 103 is particularly nice. A gel like Onzow Zerodust is also essential. So is an antistatic brush to clean record surface.
8) adjusting your VTA - one time exercise, very useful. You may need some large wrenches depending on your arm model, but totally worth it. Use a 150 gms record to do VTA adjustment.