Generic answer, applies to any machine of a similar design in that condition:
If you can, check the power supplies (not just for the machine preferably, on anything you are attempting to connect to it too!) with a load that you would not mind or manage to destroy (eg a set of power resistors). Do not test them completely unloaded!
If you are proficient enough with electricity to do so safely, wire a seriously high wattage lamp (incandescent/halogen and a few 100W. NO LED/CFL/whatever!) in series with the PSU input when first testing it. This will limit the damage that can happen in case something shorts out. Be skeptical of advice to use a variac - the kind of power supplies used in these machines are more likely to end up very stressed than soft started when you give them undervoltage.
Check the interior for foreign, conductive objects (do not open the monitor unless you are really comfortable inside CRT monitors. If you have to ask, you aren't.)
...or leaked batteries/capacitors of any kind
Connect only a minimum set of peripherals on the first run.
If you can, don't use the only copy of some disk in a floppy drive of unknown condition. Also make sure there is no unwanted foreign object or debris in the drive, this could damage floppy disks easily.
Avoid involving hardware parts with teardrop-shaped tantalum capacitors in the first test runs as much as you can, they are potential sources of serious trouble.
On the Variac practice: Often recommended with linear power supplies, to reform electrolytic capacitors and limit inrush current that could cook a dis-formed capacitor before it has time to form and damage your rectifier. Actually, it WOULD make sense with switchmode power supplies like the Apple II have IF you would cleanly disconnect everything downstream of the primary rectifier and filter circuit. Since that would mean temporarily modifying a very dangerous area in a power supply... Also, I'd advise against slowly LOWERING the voltage with a variac in any case even on a linear power supply. That would eventually cause the pass transistors to fully open and let a lot of noise onto your supply rails and/or cause uneven undervoltage in the system - risking latchup effects and/or corruption of any writable permanent storage/memory if any is connected.
On "always connect a load": Especially true on earlier Apple II models, some have a power supply of a design (a feedback-less PSMPS ... what could possibly go wrong.) that will give nonsensical voltage readings over even take damage if not loaded.
Oh, and ... if a switchmode power supply seems to be completely dead, DO NOT disassemble it unless you know what you are doing. The worst failure mode (everything downstream of the rectifier/filter is open circuit, no bleeder resistors installed) could give you 400 volts worth of welcome weeks after being disconnected from the mains.