There are some tests you can do with a partially working scope, but you are going to be very limited.

1) assuming horizontal sweep problems, you use an external ramp or the "other" sweep if you have a dual sweep model.

2) if you have a horizontal amplifier problem, using the scope in XY mode, even with the calibrator, can give you some idea of whether or not it's the sweep or the amplifier.

3) if you have a dual channel scope, you can use a working channel to debug the separate part of the vertical amplifier

These generally end up being go/no-go tests and are somewhat crude.

Function generators can make a very crude horizontal sweep.  If they're gated, then a gated ramp might work as a sweep.

Very crude, but better than nothing.

Generally, unless you have a scope specifically designed for self diagnosis, you're better off with another, completely working, scope.

Even then, a scope designed for self diagnosis makes certain assumptions about what's working, especially once you get past the "is the CPU working...."

Harvey