Great question!

Theoretically... as long as one was a Roman citizen, one could end up Emperor. In theory. That was very rare, and only occurred during the rebellions and usurpations of the 3rd century crisis, and those usurpations were almost always very short lived.

In actuality, how did promotion in the first century Roman army work? The reality was that meritocratic elevation usually only got you so far. The average Roman citizen would enlist as a tiro and could be promoted as high as Centurio (Centurion) without having to be a member of the Ordo Equester or Equites Ordines, (the "order of the cavalrymen")which was the old landowning class of the Republic which was worth enough tax value to be deemed able to field horsemen. After about 88 BC, they were no longer explicitly liable to provide cavalry recruits, but this group basically formed the upper echelon of Roman society in the Principate as all of the Senatorial class was a part of it.

Upon reaching the rank of Primus Pilus, or first Centurion, one was inducted into the Equites Ordines, and only then was eligible for promotion to higher, Senatorial-appointed ranks like Tribunus Angusticlavus. Above that were all senatorial class-only posts, such as Tribunus Laticlavus and the Legionary commander, Legatus Legionis, as well as the various governorships.

The average Roman soldier typically served for something between 16 and 22 years in the early Principate, with the option to re-enlist afterwards. Many soldiers who re-enlisted were given the privilege of becoming Centurions due to their experience, and those who didn't became Evocati (veteran soldiers who were paid more and exempt from most mundane chores). Within that initial span, it was possible to rise all the way from a recruit to Primus Pilus.

The most notable example of this was that of Spurius Ligustinus, who was a recruit from the Sabines in 200 BC, and was promoted to Centurion of the 10th maniple of the Hastati by the 3rd year for his bravery in combat. He then re-enlisted in 195 BC, and was promoted to Centurion of the 1st maniple of the Hastati. When he re-enlisted in 191 BC, he was promoted to Centurion of the 1st maniple of the Principes, and by 188 BC he had been promoted to the 1st maniple of the Triarii. According to Livy, he then was invited by Gracchus to petition the senate for a further promotion, who granted him the rank of Primus Pilus.