It depends on how soon after the apocalypse you're trying to use it.
The GPS system is conceptually divided into three segments: the "space segment", the "control segment", and the "user segment". Other satnav systems function similarly.
The "space segment" is the satellites. You need four satellites to be visible to figure out your location; right now, you can typically see nine of them at any given time. In a post-apocalyptic scenario, nobody's replacing these satellites when they fail, but assuming they all survived whatever happened, it'll be a decade or two before you can no longer count on seeing four at any given time.
The "user segment" is your GPS receiver. You've stated that your character has one that works, so that won't be a problem.
The "control segment" is where the problem lies. The GPS system relies on knowing where the GPS satellites are to extreme precision. To do this, the satellites are tracked by the US military and updated orbits are calculated and transmitted to the satellites, which in turn transmit it to your receiver (this transmission is why a non-internet-connected GPS may take a few minutes to get an initial fix -- they're waiting for the orbit data).
This data is updated every two to six hours, and is only good for a few weeks. After that time, the accuracy of the GPS system will degrade rapidly. If it's been more than a month or two since the last update, your GPS probably can't even tell you what city you're in (or near).
Even if you're in the timeframe where the GPS orbit data is still good, you won't have WAAS data. This is data on ionospheric effects and other unpredictable accuracy reductions. It's updated every five minutes, and without it, your GPS fix will only be good to within 20 meters or so, rather than the 3-meter accuracy you're used to.