Southern part: Bishop/South Lake and southbound on JMT via Mather, Pinchot, Glen, Forester Passes to Whitney is about 100 miles., less, I think, so northern half would be longer.

If you plan to do this without a car (I have) then allow some spare (3) days for logistics on either end of your trip. You can get to Yosemite via YARTS one day, next day to Mammoth, then CRST/ESTA bus but schedule is intermittent for east side all up and down the Owens Valley. From the Owens to/from trailheads, you need a thumb or a paid ride from the list below.
This is from my personal list and I have used many of these. Some info is out of date, or operators no longer in business. Check them out. Start now. You can do this from San Fran, Sacramento, Reno, Las Vegas. I have done it from them all ( "planes, trains, and automobiles") but it takes some figuring. The hardest is at the southern end, where we took the CRST bus to China Lake and rented a car one way to Vegas, otherwise you can do it from/to any of those without a car and enough time.

Shuttle/taxi: -
Big Pine Country Corners 760-872-3951 or 1-877-656-0756
Bishop Dial-a-Ride 760-872-1901
Bishop. Wilsons Eastside Sports. 760-873-7520 staff can advise
CRST/ESTA bus 800-922-1930
(760) 499-5040 info.

High Sierra Transport 760-258-6060? Bishop
Mammoth Shuttle System 760-934-3030
Mammoth Sierra Express 760-937-TAXI (8294) Scottie - used him 2009
Mammoth Cabs 760-924-2227
Ridgecrest Yellow Cab 760-384-2424
Ridgecrest Dollar rental car 760-446-4554

Taxi /Shuttle Eastern Sierra
Bob Ennis 760-876-1915 Whitney Shuttle
Richard/Bob 760-876-4443
Robert Atlee 760-873-5838
Dave Sheldon 760-876-8232 Lone Pine- used him 2010
John Pennington760-876-4545 Lone Pine
Terry 760 -876-0085
Scottie 760- 937-8294 Sierra Express/Mammoth taxi **used him 2009

drop your rig? you mean leave your car parked at a trailhead for a long time? Yes. plenty of other people and cars around on the main trailheads. On times I had a car parked, no 2-legged problems for weeks, but beware the rules for 4-legged problems. Trip #20 coming up this summer.
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SEE THE WHITNEY ZONE'S LINK TO MOST EVERYTHING YOU NEED****

Orientation notes for Whitney first timers