After Bruce escaped from the prison, how did he manage to get into Gotham? He had no access to his Batman-utilities.
You don't actually know that. I, for one, believe he had some.
Remember in "The Dark Knight", when he's in a restaurant, talking to Harvey, Rachel, and Natascha (a prima ballerina for the Moscow Ballet)?
This means that the Wayne Manor is outside of the city. Now, he obviously couldn't come in as the Batman, since he needed to pull up a little performance to get to heavily guarded Lucius Fox, but not using the suit does not mean he couldn't use his gadgets.
How exactly did he get in, I'm afraid will have to remain unanswered. The Tumbler would probably be a bit to much. The Batpod is more realistic option, but still hard to sneak by. We could assume that he has a secret tunnel, but I think that too is a bit too much of a stretch, given that no hint of such a tunnel was ever given, and there were few situations when it might come in handy.
Personally, I'd go by "sneaking on foot". After all, he is a ninja, and has mastered their art of invisibility. He might have used the same grappling hook as he did in Hong Kong in "The Dark Knight", or he could've just climb under the one bridge that was not collapsed.
I also think that not too much though should be spent on this. Ultimately, he is Bruce Wayne, and these seemingly impossible things come easily to him. For example, finding Selina Kyle and saving her from Bane and his thugs on the roof of Daggett's mansion, or finding Rachel at just the right moment in the "Batman Begins" (the subway scene),... Plenty of perfect timings with very thin explanations (or none at all). Not to mention fixing his broken back.
After all, these movies are comic adaptations, and as such they do keep a healthy dose of non-realism and cartoon physics. This way they remain lighthearted fun, not burdening us with the technical details for the sake of unnecessary realism.