Mass transit really isn't so bad. I just tried Google and it doesn't seem to know what's available.
Take the AirTrain from JFK airport to Jamaica station. This costs $8.25 now, the ride takes 5-15 minutes, depending on which terminal you start at, and you may wait up to 10 minutes for the train.
Take the Long Island Railroad (LIRR) train from Jamaica station to Grand Central Terminal. The ride takes 19 minutes, and you may wait up to 15 minutes for the train, or even longer at night. The subwy-LIRR transfer in Jamaica is very fast. The fare varies depending on time of day: morning peak is the most expensive at $12.50, at other times you should get a CityTicket for $5.
Take the MetroNorth train from Grand Central terminal to Yonkers (YNY). The ride takes 33 minutes, and you may wait up to 30 minutes for the train, or even longer at night. The fare is $11.50 during afternon rush hours, cheaper at other times. The LIRR-MetroNorth transfer in GCT involves a long ride up an esclator and can take 10+ minutes.
Once you're in Yonkers, there are several local bus lines, or you can take a taxi.
You can see schedules for both the LIRR and the MetroNorth on the Metropolitan Transportation Authority's web site: https://new.mta.info/schedules .
As you can see, 3 hours is too pessimistic.
Alternatively, in step 2 you can take the LIRR to NY Penn Station (NYP), rather than to Grand Central Terminal, which takes the same 19 minutes, and in step 3 take Amtrak from NYP to YNY, rather than MetroNorth, which takes 25-30 minutes. The LIRR-Amtrak transfer in NYP is much easier than the LIRR-Metronorth transfer in GCT. The main disadvantage is that Amtrak trains leave less than once per hour. You'd need to look at the Amtrak schedule and fares on http://amtrak.com to see whether a train leaves at a convenient time. Amtrak fares vary, I saw $17 when I looked.