Since you say the apartment is wired for ethernet and all the cables terminate where the ONT is, you have a few choices:
1) Move the router to where the ONT is. You can then connect any cat5 runs to other rooms dirrectly to the router.
2) Connect the cat5 run to where the router is to a LAN port on the router. Connect the TV to another cat5 run. Use a RJ45 coupler to connec the two cat5 runs together. This only works for one other room.
3) Install a switch at the ONT. Connect one cat5 run to a LAN port on the router. Connect the other end to the switch. Connect the cat5 run to the TV to the switch.