Yes, if a router is capable of handling Bonjour (most modern routers are), AirPrint should work. An AirPrint-enabled printer and the computer must be connected to the same network (subnet). A travel router can act as a Wi-Fi hotspot (WLAN). Some travel routers/personal hotspots provide a built-in mobile broadband (4G or 5G) modem for the Internet. Others (like the two aforementioned models) can use an external Internet (WAN) connection, if necessary. If you just want to print (locally, via Wi-Fi), the Internet would not be needed.
The printers that work via a cable (USB) are still there. The problem is that USB printers require drivers, and many printer manufacturers do not provide drivers that are compatible with macOS. This is where AirPrint can help. No additional drivers needed (if a network connection is used).