It is possible to get a reservation at Armando from their website, but it fills up extremely quickly thanks to the attention from the Tucci episode. Reservations appear 30 days in advance at midnight Italian time, which is 6pm Eastern time, or 3pm Pacific time. I reserved a table for 4 a couple months ago right after they were available, but within a half hour, they were gone. (Pro tip: practice getting a reservation for a different day without actually pulling the trigger, just so you are primed on the day you actually want it.)
Armando is a very good restaurant, but honestly, there are plenty of other really good ones in the Centro Storico. Our most recent trip we also really liked Il Falchetto, La Pollarolla, Renato e Luisa, and Pipero. All of those are great quality and are much easier to get into than Armando.