This (SSH) is possible with configuring port forwarding on the router, but make sure to set up something like fail2ban as I describe in the link above.

I've set up a Real VNC account. The client comes pre installed on the Raspberry Pi and an account is free for home use. It's been tested using my mobile for one network and my home network for another and I can remotely access RPi without issue and without needing to change the router settings.