Would have been cheaper to buy and install a RJ45 cable, even if it was a temporary so semi-pro installation. Would have been working flawless and perfect from day one but you might not have that option.
You should also have bought the H/W R3 of dap-1360 the one you have is R2, the even older one was R1 and I have been using the R3 as a "wireless client" and a standard stationary computer for Internet access, works great and only sometimes looses the connection.
In the DAP-1360 you can use "site survey" and see what other WLANs are in the area, what channels they are using and so on. Select the best one for you, 1,6 or 11 for standard 2.4Ghz networks. And configure the unit so use only 20Mhz channel width, use only WPA2/AES as it is faster, and you can stay with b/g/n without any problems. Wireless typ should be "infrastructure".
If you know your other routers IP like 192.168.X.X or what you are using. Set up the "watchdog" feature to ping that IP every 2min this should reboot the router if it fails and reconnect again. This could prevent the unit from going down a long time, as there are errors and problems with the firmwares on most WLAN units regardless of brand. Some more than others.
As you are using the DAP-1360 to connect to the other router, you should not change anything, not needed however you could check the firmware and settings of the router, so it is ok. Could be some setting in it that does create problems for your DAP-1360 and using static DHCP is a good thing if possible. Makes if more easy to configure systems and to know what IP units have if you need to connect to them. You set that option up in your main router.