The upgrades "sold" on the AGR site are 1 hour space available upgrades. If the class you want is sold out before that 1 hour mark, then no upgrade. You should also note that the upgrade coupons are only good for Business Class when coach class is the norm, and first class on Acela where the rest of the train is BC. The coupons are not valid for an upgrade to a sleeper.
While it is possible to do the upgrade by phone, frankly it's a real hassle to do so and all to often you'll hit an agent who doesn't know how to handle it and screws things up. So your best bet is to simply show up at the station just slightly before the 1 hour mark and get in line at the 1 hour mark, or perhaps a few minutes before if there is a long line. At your option you can either present a ticket that you've already obtained or you can simply give the agent your reservation number. Either way, make sure that you have your printed confirmation with you just in case that agent screws things up and allows the computer to reprice the ticket to the current bucket level. Your printed confirmation is the only proof you have that you paid X originally. If the computer reprices and the agent didn't pay attention, the agent cannot see what the old price was in the computer.
You must do the upgrade with an agent at the station eventually, even if you called initially to start the process and hold the seat because you couldn't get to the station before the 1 hour mark. The conductor cannot accept the coupon on board along with your lower class ticket and will either ask you to pay for the upgrade or move to the correct car that you have paid for.