I don’t really understand the vatspy software. How can I determine if there are vatsim controllers available for me to fly from one airport to the next with Ground, Tower, Departure, Center, Approach, Tower… services the whole way? And how long will they be on? Each time I’ve logged on recently the controller announced he was closing in 5 minutes.
VATSpy, SimAware, VATGlasses, VATSIM Radar, and all the other map tools show you two things: who is online right now and in the case of ATC (some more, some less accurately) what area they cover. Nothing more, nothing less. I think some may have functions to show predicted coverage based on past coverage and bookings as well, but those obviously aren’t a guarantee for staffing.
Short answer: you can’t.
Long answer: you can check for events and bookings to get a rough idea of when there will be coverage somewhere, but those are also no guarantee for staffing. Everyone is a volunteer after all.
Some put an expected logoff time in their controller info, but most don’t. Usually, it is a pretty spontaneous decision, too. Unless a controller has a subsequent appointment, they usually don’t go into their session thinking “I’ll go offline at this and that time”, they will make that decision based on traffic, surrounding staffing, how much fun they have (that last one is usually highly dependent on pilot competency), etc.
However, the longer a controller has been online, the more likely are they to go offline soon-ish. Similarly, the controller’s local time (usually the same as the airport’s local time) also plays a role: the later in the day, the more likely they are to go offline soon. So if you see an online controller that has been online for five hours already and the local time is almost midnight, chances are high that they will close very soon.
All that said, however, some general advice: don’t chase ATC. If you plan your flights based around ATC staffing, you will in almost all cases be disappointed because someone goes offline before you arrive in their airspace. Try to plan flights that interest you and if there’s ATC online somewhere along your route, see it as a bonus 
…and just add - check the VATSIM event page. You can find many weekly events, where atc is garantied to some extent.
VATSpy map also doesn’t accurately depict VATSIM coverage areas.
For example, VATSIM ZNY covers KPHL; in real life, KPHL underlies ZDC airspace (see an IFR low chart for this example). Not sure why VATSIM is set up this way. Just another sim quirk to keep in mind.