Yep, same exact Rome as modern day Rome. Although there are Romans who aren't from Rome that wouldn't happen for centuries after what you are referring to.

They got there by land and sea. I think it would be helpful to look at a map of the Roman Empire at its height for a sense of context.

https://s-media-cache-ak0.pinimg.com/originals/37/6c/d6/376cd6f79ad5da08be69cad05ad72cec.jpg

The amount of control they had over the government varied in the years before and after the birth and life of Christ. Sometimes they had what were essentially client kings who ruled the province with the approval of Rome. At other times, they exerted direct control over the province. During the time of the crucifixion of Jesus (assuming this as a historical fact for the sake of argument others would be better at debating that issue) the Roman Prefect who controlled the area was Pontius Pilate.

As to the why? Judea first came under Roman influence, via client kings, around 63 BC. At this point, Pompey the Great was nearby fighting the tail end of a war against the king of the kingdom of Pontus. And he intervened in a local civil war to establish Roman dominance. Judea remained important to Rome because it was both a land and sea route to Egypt, their most profitable province.