If you don't already have a programmable thermostat, you can get one for about $35-50 at Lowe's or Home Depot. I bought one for our last rental home (it was a huge house, not an apartment) and it was very straightforward to install. Literally just pulled the old thermostat off, disconnected the wires and plugged them into the color coded spots on the new thermostat.
No need to spend $300 on a Nest. A basic programmable digital thermostat will save you money in the same way.