Time/price of driving depends on whether you are driving in during peak or off peak hours, and whether or not you have an EZ Pass. (If you don't have an EZ Pass, I wouldn't drive at all.)
Assuming you DO have an EZ Pass, and you will be driving during normal peak hours (weekdays, 6-10 AM and 4-8 PM), the toll costs will be $10.25 for the GWB and $5.33 for the Whitestone Bridge. Thus:
$10.25 + (2 x $5.33) = $20.91 total tolls per day. (Plus the cost of gas.)
If you commute during off peak hours it will save you $2 on the GWB.
Tolls - Bridges & Tunnels - The Port Authority of NY & NJ
mta.info | MTA Bridges and Tunnels' Toll Cars
As far as using public transportation, I have heard that the NJ Transit bus to the
Port Authority Bus Station takes forever (like an hour+). Not sure about the gypsy buses, though. Taking a NJ Transit (or gypsy) bus to the
George Washington Bridge Bus Station on 179th Street in Washington Heights is much quicker (it's also about a buck cheaper than the NJ Transit bus to the PA bus station). From there, you can take the A train down to 42nd Street and then take the 7 train out to Queens. Commuting by bus + subway will cost in the $10 range ($5 for subway, plus $5 or $6 for the bus fare if you get a monthly pass).
Time wise, whether driving or taking the bus + subway, I'd bank on about 1 hour, conservatively. I don't think there would be a significant time difference either way.
Personally, I would drive from Fort Lee. But I am the kind of person that would rather deal with traffic than with crowded buses and subways that are also slow.