New Army Pass probably has a little snow left on it, but it should be pretty easy to climb and descend. Climbing Whitney from Horseshoe Meadows and returning that way is usually a 5-day backpack.
If you're main goal is Mt Whitney, check for permits to climb the Main Mt Whitney Trail.
Here's the link:
https://www.recreation.gov/permits/233260 Check it multiple times per day, and no-shows and cancellations should show up. There is currently an overnight permit available starting two days from now... (actually less than 30 hours from now).