In SPQR, Mary Beard mentions that if Tiberius were to find himself emperor at the time of Commodus, he would be able to rule pretty comfortably. However, only a few decades later, he would have no idea what it meant to be emperor of Rome. How true is this claim?

It is a pretty true claim. The role of Emperor was pretty similar in scope from Augustus (starting in 27 BCE) to the end of Commodus (192 CE). That era is commonly referred to as the "Principate" era of the role of Emperor in that the Emperor was regarded as the first citizen. Throughout that entire time period the Emperor was most powerful, but the Senate still had some role and Rome was still the most important city in the Empire.

While scholars mark the Principate Era as extending until Diocletian in 284 CE, the Severan Dynasty (ruled 193 to 235 CE) began a change in the make-up of the Imperial power structure. Then the Crisis of the Third Century (approximately 235 to 284 CE) necessitated change and finally Diocletian (284 to 305 CE) instituted the formal change to the Dominate.

Emperor Septimius Severus started the Severan Dynasty and he lived by his deathbed advice to his sons: "Be harmonious, enrich the soldiers, scorn all other men." The scorning included the Senate and anyone who was not a soldier. This shifted the power dynamics to the Emperor still being most powerful, but the Legions being the basis of his power.

The enrich the soldiers also led to vast changes. Soon the common soldiers and their commanders expected to be paid off on a regular basis. If they did not receive their money they were known to turn to mutiny fairly quickly. This would have been a change for Tiberius. Sure, he paid his soldiers. Augustus left a nice donative to the Praetorian Guard in his will and Tiberius provided another for their loyalty during the Sejanus incident, but he would have been surprised by just how much of his power now relied on basically bribing his soldiers for his power.

The Crisis of the Third Century also would have brought in a huge change. The Pax Romana (Roman Peace), which had begun in the reign of Augustus and lasted for 200 years was not dead, but buried deep in the ground.

Emperors were expected to be everywhere and were constantly putting out fires. This would have been vastly different from the days of Tiberius, where he basically never had to leave Italy, and the days of Commodus, where he was allowed the freedom to pretend to be a gladiator. Emperors had to be on the road constantly, Italy and Rome started to slip in importance and their grand stature as the Emperor had to focus outside the home province.

This all culminated in the reign of Diocletian. Diocletian was able to go twenty years without visiting Rome and when he finally did he hated the home city. Strategically speaking, Rome was too far removed from the rest of the Empire for an Emperor to be stationed. The new circumstances of the Empire called for a more active Emperor and one who could respond quickly to new challenges as they arose. An Emperor would be quickly ousted from power if he tried to pull a seclusion act in Capri, which obviously Tiberius got away with. Commodus got away with even more than a seclusion.

I hope this gives a little more context and kind of answered your question. If you want any clarification let me know.