***How long can I postpone the repair? ***
Until the timing belt breaks. Which may be tomorrow or might be 2017
Is there a way to tell if the timing belt is worn or about to go?
Yes. It will break. (The non-fippant answer is "No")
What happens if it breaks while I am on I95?
Your vehicle will coast to a stop and will not start again until you get it repaired. With towing charges and the "how are you going to take the work elsewhere" factor, the repair will probably cost more than your mechanic is asking and may be several thousands of dollars more if you have(had) an interference engine before you destroyed it.
What is a reasonable price for the repair?
If you replace it now $500 to maybe $1400. Depends on the vehicle, labor rates, and whether the water pump is replaced as well. Replacing the water pump when the timing belt is replaced is generally recommended, but strictly speaking, it's optional.
Any other suggestions?
If you have tools, a place to work, and know how to use the tools, you may be able to do the job yourself. It is a fairly difficult job and not one I would recommend as a first automotive project. Other than that, you could/should get bids from several mechanics. Or depending on your situation, you could leave the car in DC and depend on the Metro, the NY subway and the Chinatown Bus (a cheap bus service between several NE corridor cities) for transportation.