Assuming that RMN enlisted (or was drafted) either before or immediately after Pearl Harbor, the best way for him to become an Army officer was to be selected for Officers Candidate School after basic training or after a short time as an enlisted man.
Agreed. OCS is the best method. In fact, as a college graduate and practicing attorney, he would probably be selected as a potential officer right at the start. There is very little chance the Army is going to keep a college graduate as an enlisted man when there is a desperate need for junior officers. He'd be given a commission as a second lieutenant. Most likely he'd rise to the rank of Captain as AH suggested. However, if he served in combat, he'd potentially receive commendations or medals that burnish his reputation.
In order for him to be promoted to major or above, he'd need to have lots of combat experience and have performed very well. As well as getting lucky in the opportunities for promotion. Major would be pushing it. I'd only see Colonel if his service was completely outstanding AND a senior general noticed him and choose to promote him to staff duty near the end of the war.
The chances of Nixon befriending Eisenhower is extremely low. However, if Nixon ends up on some general's staff, then Ike would learn of the kind of man Nixon was through knowing that general. Certainly it would create some kind of bond if they had a mutual friend (Ike's subordinate and Nixon's superior) when their political careers intersected. Assuming OTL's Nixon's career in politics, Ike might be much more strongly committed to Nixon as VP.
However, if Nixon is at Monte Cassino, it means he probably stays in the Italian theater. That makes it much less likely he comes into any kind of contact with Ike. Why do you want him to be at Monte Cassino? The best chance is if he is in a unit that gets transferred to Britain for the invasion of France. I wonder if it would be possible for Nixon to be posted in the 1st Infantry Division? While he might not make it in time for Torch, perhaps he'd be a replacement for the Sicilian campaign. If he was in one of the units that received heavy casualties on Omaha Beach and displayed great leadership and heroism, it might be enough to fast track him into available officer positions. Let's say he was one of the men who helped Colonel George Taylor, and commander of the 16th Infantry Regiment, rally. Taylor was impressed and remembers Nixon. Later when Taylor was promoted to Brigadier General and later assistance division commander Taylor eventually brings Nixon on board as staff after Nixon continues to display superb combat leadership in the months following Normandy.