I'm buying a RB2011UiAS-RM and I was hoping to cover my house with Wifi with a single POE Access Point using the POE port on the router. Does anyone have a suggestion for a relatively inexpensive POE Wireless AC Access Point?

I don't know if just one of them will cover your entire house, but Mikrotik sells these cheap indoor units, that can be setup as AP:

MIPSBE - 802.11ac single chain
https://mikrotik.com/product/RB952Ui-5ac2nD
https://mikrotik.com/product/wsap_ac_lite

ARM - 802.11ac dual chain
https://mikrotik.com/product/cap_ac
https://mikrotik.com/product/hap_ac2

All of them are 2.4GHz dual chain too.

By the way: unless You know what You are doing, don't buy an RB2011: they have a very weak CPU. The two ARM ones I posted before will run circles around an RB2011, speed and routing capacity wise.

With what You would spend with one RB2011 you could buy 2 ARM ones. That's enough to use one as router, two as AP and spread the signal better.

By the way: unless You know what You are doing, don't buy an RB2011: they have a very weak CPU. The two ARM ones I posted before will run circles around an RB2011, speed and routing capacity wise.

With what You would spend with one RB2011 you could buy 2 ARM ones. That's enough to use one as router, two as AP and spread the signal better.

Agreed. The 2011 has been around a very long time. It should be end of sale/end of life.

I ran an RB951 for years and was happy with it; then replaced it with a hAP ac lite I got as checkin swag at a MUM, which added 5GHz. I'm equally happy with that. I ran both from a central location, wall-mounted, and fed with POE.

Realize that if you have a large house or many intervening walls, running a "stronger" AP doesn't solve your problem, because it doesn't also make your devices "stronger."