A drill would be a good investment, lightweight general purpose cordless drills are available at a surprisingly reasonable cost.
The joint compound plan might work alright. You don't state how big the original hole was; if it is large (like, larger than the new thermostat) then you might have a tough time patching without something behind to "back up" the patch. Consider bridging the hole with a strip of stiff material, like a strip of drywall or even a wood paint stir-stick. You would slide the strip into and beyond the edge of the hole on one side, then pull it back so the strip overhangs both sides of the hole on the inside (I usually put a screw through the strip in the middle, to serve as a temporary handle). Run a screw through the wall and into/through the strip on both the right and the left, to pull the strip up against the back of the wall/hole on the inside, then remove your makeshift "handle". The attachment screws would have to be driven below the wall surface so you can patch the screw holes at the same time you fill the main hole with joint compound. Smooth and sand, then use plastic anchors to secure your new thermostat.