Here's what I have done multiple times when running a cable was not feasable, but wanted almost the same performance as running a cable. Use powerline adapters. If you electrical wiring is good, TPLink powerline adapters will work very well at connecting the access points with a wired setup. A transmitter powerline adapter near the main network/router with it connected to the network via ethernet cable... the powerline receiver at any end of the house... sync them up... plug in the distant Ubiquiti AP to it via ethernet cable... and then set them up as a mesh in the Ubiquiti app on your phone/tablet.
It won't be as 100% as running an ethernet cable end to end between each AP and a network switch, BUT it will be at least 85-95% if not more. BUT that's still a lot better than just a wireless mesh. Ubiquiti APs can be set up as a mesh via...