Your idea is perfectly feasible and Zeroshell is THE tool for you:
Web login before surfing is possible using the Zeroshell’s embedded Captive Portal.
No additional components will be necessary considering the old PC has multiple Ethernet cards (at least 2, one for accessing the WAN and the other for the LAN.)
You’ll probably just need a switch so you can connect the 2 APs on the LAN.
NOTE: The Captive Portal is part of Zeroshell
Good luck with your project!