Hi,Laura
White Mountain is "about" a 1 hour drive from Lone Pine to Shulman Grove -- home of the bristlecone pine forest. The loop trail is approximately 10,000 feet in elevation. I believe the trail is 4 miles RT.
The drive to Schulman to Barcroft gate is another story requiring high clearance vehicles (some low clearance have made it). Can make for a very bumpy ride.
If this is a day before your scheduled MW hike, I would take it easy and hike the grove. Bring plenty of water, snacks, sunscreen, jacket. Don't orget to bring that camera. The one tree you are taking a picture of could be
Methuselah!
Have fun.