An old scope like that may not be worth a lot of investment of time or money, although I understand the attachment one develops for old, familiar, instruments. Is this an instrument you have some history with, or was it newly acquired?
Checking the capacitors is a good start. If you lack a capacitor checker, remember that some DVMs come with a capacitor check function and they are often quite inexpensive. Alternatively, some crude tests are possible with just a DVM or VOM. Having several identical capacitors for comparison greatly facilitates such cheap and dirty tests.
Set your DVM/VOM on a range that charges the capacitor over a perceptible period of time. Apply your tests leads to the capacitor and observe the dynamic response of the pointer. (It is much more difficult to to this test with a DVM because looking for a trend in a series of flashing numbers is difficult. Additionally, the current supplied by the DVM in the resistance mode may be so low that the capacitor does not charge up in a reasonable period of time.)
At the instant you apply the leads to the capacitor, you should see the Ohmmeter pointer dip to nearly zero, then gradually climb back up to near infinity. If the voltage indication does not climb up to a very high resistance, the leakage is probably too high. If the pointer doesn't dip to near zero initially, the capacitor may be open. If the needle dips to a low value and stays there, the capacitor is probably shorted.
You are lucky that you have several of each type of capacitor on the board. This allow you to observe the typical charging time of each capacitor of interest and compare the needle dip characterists of each.
Better capacitor tests can be performed with higher voltages and more sophisticated equipment, but the tests described above can tell you most of what you want to know about your voltage multiplier capacitors. If you do use high voltages to test the capacitors, be very careful.
Since you say that you heard a buzz just before the 'scope failed, you should test all rectifiers in the power multiplier circuit and other parts of the power supply. The test is most reliable if you disconnect one end of the diode from the board to avoid influence from nearby components. You should see a low resistance (<100 ohms) in the forward direction and a very high resistance (tens or hundreds of megohms) in the reverse direction. Here, again, testing at a few volts out of the ohmmeter is not as revealing as testing near diode reverse voltage rating, but it is readily available, cheap, and capable of revealing ordinary defects.
Don't overlook simple, very close examination of the circuit board for signs of overheated components, bad solder joints, cracked conductors, etc. A magnifying glass is very helpful here.
Be sure that the unblanking pulse is getting to the proper pin on the CRT.
Check the CRT filament pins for continuity. If the tube filaments are shot, you are pretty much up the creek. Unless you stumbled across a replacement CRT at very low price, it will probably not be worth replacing the CRT, even if you can find one.
Use a high voltage probe to confirm the presence of the proper high voltages on the CRT. You will not be able to measure the exact voltages due to loading by the measuring probe. However, you should be able to confirm the presence of a high voltage at the HV terminals.
Check for the presence of appropriate voltages on the power transformer. Old 'scope power transformers are usually specials and not worth the cost of replacement.
Have fun, and be careful.
awright