OK, Lee is screwed. There is no way he is beating Grant so the war end in late 1863 or early 1864 I guess.
Grant is in command of the Union Army of Virginia during the Northern Virginia Campaign then in late summer 1862.
Grant will be similar to Pope. His promotion is likely to be resented, and he doesn't have the same reputation that he did in 1864. Still, victories at Donelson and Shiloh will be worth something. He also has better temperament and talents than Pope.
Grant will have the same objectives as Pope: 1) defend Washington and the Shenandoah Valley, and 2) draw Lee away from McClellan and attack him.
Lee always tried to understand the mind of his opposing general. Likely he will be respectful of Grant's own success, and Lee doesn't have the reputation yet that he earned IOTL. He will still determine McClellan won't attack, and will therefore try to destroy Grant's Army of Virginia in detail.
Lee will likely be aggressive in tactics attack Grant, but probably won't risk dividing his forces. Most likely we'll get stalemate where Grant stays on the field after an inconclusive battle. He waits for McClellan's Army of the Potomac to arrive, and together they will pursue Lee.
Grant and McClellan will likely not get along, and Lee will try to exploit that. Sensing Grant is the better officer, he will likely try to destroy McClellan next, but McClellan's own paranoia and caution will likely prevent it.
Face with two armies in the field, Lee is likely pushed south with the possibility of Richmond falling before end of 1862. Presumably, when Richmond falls, Lincoln issues the Emancipation Proclamation.
Assuming Grant does well, Lincoln promotes him to theater command. He either combines both armies against Lee to lead himself, or he is in the same role in 1864/1865 IOTL. Lincoln probably kicks McClellan upstairs as a training and organizing role, but without field command and without influence on strategy.
When 1863 begins, the Confederacy will be in a bad position with a provisional capital elsewhere - maybe Atlanta. War is probably over before year's end as the Mississippi River is finally taken, and Grant presses south from Virginia into North Carolina. With the Confederacy split in two, and not having the resources of Virginia, it'll be tough.
The war could end with a siege of Atlanta as Grant comes form the north while Thomas/Sherman comes from the west.