Ubiquiti is the cheap solution. By far the best bang for your buck in this scenario. With the solution above, you should be able to keep it under $500 total.

Search google and youtube for "wireless bridge" or "wireless bridge to connect two buildings" and you should be able to get most of the info you need as well as a decent shopping list for Amazon.

For reference two of these would work fine for the bridge, and one of these as your access point for clients on the remote end. Set up the SSID and password the same as your wifi in the house and your devices should roam between the two without much hassle. Beyond that, you just need ethernet cable to connect everything.