Yes, ARD will work, but the issues involved in configuring the routers and dealing with often-shifting IP addresses can make it a headache. I'd suggest you consider instead one of the server-based remote control solutions such as LogMeIn which don't have most of those issues. I really don't think you want to try and get your mom to deal with router settings and reporting the IP address to you on a regular basis.
Yes ARD will work over the internet. My current config has a mac at home that I can access remotely. I also manage my mom's mac remotely - she has DHCP ip from service provide, a quick script on her mac (auto running) updates Dynamic DNS free DNS freedns.afraid.org so I can just use "mom.homesystem.org" (example) and not worry about the IP address etc.
There are several guides that can be found via searching google etc.
Set up the remote computer with a fixed IP address
Set up ARD
Open 5900 and 3283
Use a web page like http://www.whatsmyip.org/ to have your mom tell you her IP.
The guides I alude to above will detail all the bits needed and any config. ONCE its setup she won't have to change any settings on the router, nor report the IP address.