Using the older rail alignment along the scenic Rhine River, Frankfurt Airport to Bonn takes about 2 hours, with the best available class of trains being the IC (Intercity; good trains with 1st and 2nd class and usually food service). This would take you directly to the Bonn HBF in the center of town (main station) without a transfer.

The new high-speed alignment bypasses the Rhine Valley and goes to Cologne in about 1 hour. ICEs use this route. As pointed out before, Bonn would be accessed from a secondary station like Siegburg, or you could go to Cologne and change trains for the 20-minute trip to Bonn.

All of these will leave from the easily-accessed newer long distance station at Frankfurt Airport (Fernbahnhof) rather than the older station that now just handles local trains like the S-Bahns.

For convenient access to Bonn I would prefer the scenic trip along the classic Rhine River on the older alignment.