Sorry Mary, but you can not walk to your address in 15 minutes. Google Maps says 33 minutes and i think that is optimistic given the number of street crossings and crosswalk lights.
EDIT: David is usually right when it comes to Budapest so maybe he thinks the Vienna train stops at Nyugati station. That would be closer. But then, maybe I am guessing wrong too. The MAV and OBB trains stop at Keleti, but the RegionJet discount trains I believe stop at Deli station (which is not convenient at all). So check your ticket and let me know if we have to rewrok this.
I would get off the train, walk to the center front of the station. Walk down the stairs and go to the BKK office and buy a paper 72 hour TravelCard (EDIT: Sorry, you are over 65. You dont have to buy anything). Then follow the signs to the M2 Metro line. Board it in the direction of Déli pályaudvar (Deli Train Station).
Now you have two choices.
FIRST Stay on the M2 for 3 stops to Deák Ferenc tér where you change to the M1 (townards Mexikói út) and get off at the second stop the Opera Metro Station. Your AirBnb is maybe a 4 minute walk behind the Opera house.
SECOND Stay on the M2 for 1 stop and get off at the Blaha Lujza tér metro station. Then go up the stairs to the street and board the 4/6 tram in the direction of Széll Kálmán tér (moving right to left from the top of the metro stairs) Sit on the right side, the view is interesting as you pass the NY Palace and a number of other wonderful old buildings. Go 3 stops to the Oktogon. From the Oktogon you can walk something less than 10 minutes down Andrassy ut (beautiful) or you can take the M1 Metro 1 stop in the direction of Vorosomarty ter and get off at the Opera stop.
The first way is efficient, the second lets you see some of Budapest. I would do the second option, but I am comfortable in town. With luggage the first is probably easier as the 4/6 tram gets croweded at times.
For your departure, Call City Taxi the day prior and request a pick up at 14:00. It will take not less than 30 minutes and you are coming up on rush hour so maybe a tad longer. Worst case you will arrive a 14:45. Its an easy airport so 2:15 is early enough. If you worry (like I do a lot), then call theTaxi for 13:45 pickup.