Does the Asus rt-n66u require a modem to connect to the Internet?
I don't have that model, but generally, yes. Some routers are configured as repeaters and get their signal from another router (which through the WAN port is connected to a modem). manuals are usually available for download at the manufacturerer's website in the support section.
The RT series doesn't have a built in modem, so if the type of internet connection you're using requires a modem (many do) then you won't be able to connect without an external modem.
Asus do equivalent or better models in their other ranges that have built in modems of various types, like the DSL-N66U and 4G-AC66U