1) Yes, you can take the San Martín train (a bit of an adventure), get off at the Palomar station and take a taxi to the airport (I personally wouldn't walk there). Its about a 5 minute taxi ride. There are two taxi stands right outside of the station.
The San Martín train ride from Retiro is 50 minutes, and about 35-40 from Palermo. Give yourself plenty of time since the train is often delayed.
2) Another option is to take the Urquiza train (much nicer, safer, cleaner and more reliable than the San Martín). Take the train from the Lemos station in Chacarita, get off at Martín Coronado and take a taxi (there is a stand behind the station) to Palomar. The train ride will be about 30-35 minutes and the taxi will take another 15 minutes. This would be the least stressful option, in my opinion.
3) If for some reason you can't take either train, the 123 bus passes by the Chacarita station and goes to Palomar, BUT it will be a 1-2 hour ride.
If you have any other questions, let me know. I know the area quite well.
EDIT: See update below. Taxi station is right near train station.