Please help me with temperature alarm system
If the modem is the only problem, replacing it might well solve the problem. As for finding a particular model of old modem at a non-outrageous price - ebay; sometime patience is needed for a very specific model or to get it at a non-absurd price, but generally modems are pretty cheap there. 

Where the number is programmed - to be certain, you'd need a serial cable splitter so you can see what's going to the modem (connecting a computer so you can view the traffic between the modem and the controller.) Most likley it's in the controller, and a dialing command is being sent to the modem, but some fancier models of modem did have number-storage abilities, if I dimly recall the state of 15 years ago correctly. 

You might very well find that you will be better able to solve the problem with a complete replacement using newer (and currently documented) hardware (telephone dialing temperature alarm or similar for a product search) rather than patching together old equipment - these things become obsolete, and one way to detect that is the absurdly high charge to even look at it.