Likely you will need MORE than a gallon of antifreeze but that is pretty much a standard way to winterize when no winterizing valve is installed. You could install one near the water pump for about 15.00 and cut down on your antifreeze use by quite a bit.
As for pushing the check valve in the pressure water connection, this is just fine but turn off the pump and release almost all pressure first to avoid blowing the seat on that check valve and causing issues next spring. It takes almost zero pressure for antifreeze to seep out the check valve. Unseat it and you will pump your fresh water out the city water connection next spring whenever you turn on your water pump.