First, I would get a proper service manual and for the correct model. Although they are similar, there are differences between A, B, and non-A/B versions. If you haven't worked on these scopes it can get confusing if you're not looking at the right manual.
Unfortunately, the manual on elektrotanya.com is incomplete. It does not have schematics or placement diagrams. I wasn't able to locate one with schematics anywhere for you. However, you can buy a high quality scan from Artek:
http://artekmanuals.com/manuals/tektronix-manuals/ I've purchased manuals from them before; they are top notch.
There are differing opinions, but I would not run out and start replacing components without at least doing some basic troubleshooting. Wholesale component replacement, depending on your skill level, can sometimes make the situation much worse by compounding re-assembly errors with the original problem.
If you buy the service manual, there will be troubleshooting flowcharts that start with checking the power supply voltage levels and ripple. If you have a DMM or other working scope this should be easy. If you say this scope works after you press A/B, you might even be able to use it on itself for the simpler tests.
Do a visual inspection. Look for capacitors that looks obviously bad, such as leaking or bulged. Try gently pushing on connectors or reseating them. I'd pay particular attention to the A5 control board, which is on the right-hand side.
You should also try to get into the diagnostic routines to see if they report anything interesting. The information on how to do it is included in the elektrotanya.com manual in the maintenance section. It may not work given the issue you're seeing with the display, but it's worth a try.