It's a pretty decent risk, while the fault could certainly be display or power supply related, they're also YIG oscillator based, and there are a lot of RF bits that can be challenging to troubleshoot without RF gear. I don't mean to discourage too much, but especially if you're unfamiliar with that sort of gear or repairing it, I wouldn't put the odds of it being a simple failure and nothing more without any issues that manifest outside of what it can test on it's own (verifying output and input with external instruments). I've got a 4395A as a project currently that was sold in what appeared to be working condition but was entirely untested aside from powering up - passes all self tests, but the output is way off frequency for a chunk of the sweep which manifests as noise that can't be normalized on the trace (not to mention no actual valid measurements at some frequencies)... it's currently low in the stack of projects.
I think the odds of your perspective unit being unrepairable are pretty low, but I would expect the repair to be somewhat in depth. There are parts and assemblies from other units available, so it's not like they'd be impossible to find, but they could be expensive and there is some likelihood that you'll need to get one. Testing the monitor output is extremely easy - a <$10 ebay special VGA to usb capture device can do it sometimes, but I got a nicer VGA input LCD at my local thrift store for $15 and it does a better job (wider signal compatibility). There is a service manual available, but it's not schematic level, just assembly with block diagrams (at least, for the 4395A), and you'd have to see a lot of 4396As to make any real statement on reliability on their PSU or LCD... so I wouldn't put much stock into it.
So all in all, it's a gamble, as you probably already knew. You've got to weigh that against how much you want it, how much experience/ability you have to repair something like it, and how much you can afford (in time and cost). I think the general rule of thumb is to not gamble much and never assume it will be an easy repair, but if there's a chance of it and you're equipped to troubleshoot for the long haul, you're likely to get a working instrument out on the other side.