If you have IP connectivity from home to the router in question, (via VPN, etc), you could do this without any phone lines on either end. If the aux port on your router is like the one in this document: Cisco Description of Aux Port, then it appears you can just use it like an RS232 serial port by means of an adapter. In that case, adapt it to an RS232, and then get one of the many products that come up when you google for 'RS232 to IP Converter'. Most of these products basically act like telnet servers that you can connect to remotely and then your telnet commands get proxied back and forth to the serial device. You would just telnet into the attached converter and find yourself at the Cisco prompt.
You could also skip the converter and hook the serial console right to a PC and then remote into that PC via ssh or RDP or whatever to access it.
If for some reason, serial won't cut it and you are forced to use a modem connection, you can purchase a 2 port telephone line simulator, connect the cisco modem to one side, and an old fashioned serial modem to the other side and then use the RS232 to IP Converter on the side of the 2nd modem. Then when you telneted in, you'd have to do an 'ATDT' command to dial the number configured for the other port in the line simulator to get to a prompt.
Alternatively you could leave out the old serial modem and IP to RS232 converter and instead use a normal external USB Modem (or internal modem even) hooked to a PC, and then remote into the PC via remote desktop or ssh or whatever to control the modem to dial the aux port.