A couple things, El-Diablo
First, find out what model number of Nortel phone you have on your desk. Many of Nortel's phones are a digital line type with completely different line and ring voltages than the typical POTS or analog lines. In short, if you plug that phone line into your modem, you could smoke your modem. If you can post the model number of your phone, then I can tell you for sure. Its probably on a sticker right on the underside of the phone.
A lot depends on what line type that phone is using. If it is the digital type, then DTMF tones won't work for dialing the number. Let's just hope its not digital.
I wanted to point out that there are handset lifter products out there, usually coupled with a cordless headset solution. The good news there is that it will almost certainly work with that Nortel phone of yours and most of the Nortel phones have dedicated headset jacks that you could use.
If you can figure out how to get the handset lifted and the 81001111 number dialed, the rest is probably easily done via DTMF and can probably be done using many of the ways already suggested in this thread.
Hmmm...wonder if you could tear the phone apart, replace the buttons with transistors/relays/whatever, and have the phone under control of an MCU. Now that would be a sight to see.
Do you have a call forward capability on your phone? You could forward your work phone to 81001111 and then just call yourself from your cell phone. Now that would be super easy.