Laura, curiously enough I and my family did the Methuselah Trail at Schulman Grove this afternoon as a pre-Whitney hike. First, an absolutely glorious panorama of the valley, the Sierra, and the ranges to the east of the Whites. Most breathtaking view I've seen in the eastern Sierra.
The drive from Big Pine to Schulman Grove took about an hour up (and around 25 minutes down

), with no traffic, but several stops to admire the scenery. We drove it from Mammoth Lakes, but I'd guess another hour max for the Lone Pine to Big Pine segment, so allow 2 hours each way to be safe. CHP loves 395 . . .
The Methuselah Trail hike - the longest one at Schulman Grove - is a 4.3 mile loop beginning at 10K' and reaching about 10,800', but the cumulative elevation gain seems a helluva lot more. I swear this trail has more up than down, which is a neat trick for a loop. My wife, 10 YO daughter and I took about 4 hours to do this, stopping regularly for photos, lunch, admiring the view on the east side of the trail, etc. The Ancient Bristlecones are just amazing, and the trail itself is fun. Maybe not the best choice for a first acclimation hike right out of the chute if you're coming from the lowlands (like us), but this was our 3rd day of hiking above 10K', so everyone was well-adjusted.
I start up Whitney Wednesday, 2 days away, and this was just about all the work I would have wanted that close to permit day. Cool-out, rest day tomorrow. Last year I did the two Cottonwood trails and Mono Pass in the 4 days before Whitney, and I think I may have overdone it. I strongly discourage you from putting forth much effort/energy at least 24 hours before you start Whitney. Best of luck!