Why does he check in? .. He didn't have to go to the desk and give his passport, he could have gone to the hotel, causally walk in, and go upstairs without arousing too much suspicion.
Jason Bourne wanted to be tracked.
When he was first traveling from Tangiers to Naples, Bourne had no problem giving security his passport. And then, when his movements were discovered by the CIA, the following was said:
ZORN: Why Naples? Why now?
KURT: Could be random.
CRONIN: Maybe he's running.
ABBOTT: On his own passport?
KIM: What's he actually doing?
CRONIN: What's he doing? He's making his first mistake...
NICKY: It's not a mistake. They don't make mistakes. And they don't do random. There's always an objective, always a target. If he's in Naples, on his own passport, there's a reason.
Keep in mind that, a critical aspect of this film is the process of Pamela Landy becoming familiar with the finer details of Treadstone. Up until now, it was believed that Vladimir Neski was killed by his wife. However, by Bourne going to the same hotel where Neski died, and even to the same room, Pamela Landy is then convinced that Neski wasn't killed by his wife, but in fact, by Jason Bourne.
CRONIN: The room he checked into was across the hall -- why, why would he come here?
LANDY: I recognize this room from a photo. There was a chalk outline right here, around the body of Vladimir Neski.
CRONIN: This is where his wife killed him?
LANDY: Do you still think his wife killed him?
By Bourne keeping his travel patterns "out in the open", and, with the investigative power that Landy has (with her the Deputy Director of the CIA), he's bringing into focus the history of that hotel, and, subsequently, providing Landy with new information about Treadstone.
Making a reservation was the quickest and easiest way to get to the room. If Jason was planning on being there for only a few moments, this wouldn't be such a big deal. It is important to mention though that, had Jason actually been given the reservation for room 645, he would have been caught by the SWAT team. But, it would seem that luck was on his side, given that he was issued room 644 instead, providing him a chance to escape undetected.
It's quite possible that, because Jason had to take an elevator to get to the room (and/or possibly go through other doors), the elevator (and/or other doors in the hotel) could have required a hotel card to activate, thus requiring the reservation to be made in order to get a hotel card. This seems likely considering that it was a nicer hotel, and millionaire politicians were staying there.
At the desk he learns the room is occupied, but NOT if the people are currently in the room. He still goes to the room he wants to and picks the lock to get in, despite not knowing if the people who booked the room are there.
Jason rang the door bell before picking the lock.
When Jason first approaches the door to room 645, he rings the bell. Since nobody came to the door and he didn't hear anything, he then picked the lock.
You do have a good point though..
about Jason just walking to the hotel room without a reservation, acting like he's a guest that's already checked in. If Jason's intent was to remain hidden, he probably would have does this. In fact, Bourne takes this kind of tactic when tracking Landy while she was staying at Hilton Hotel.