Not sure exactly without seeing what you're doing, and a while since I've used premiere, but if you are doing what I think you are, the easiest way of doing it, without having sync problems, is to export a copy of the entire audio for that particular section (including the bits you don't want to alter), treat the whole thing, then import the copy and then mix between the original and treated version of the audio.

Without knowing premiere that well, and assuming you have one clip or section which contains the whole sequence including the bit somewhere in the middle where it changes, basically,

1. export a copy of the audio from that clip or alternatively, load that clip straight into audition if it lets you
2. modify it in audition (add reverb or whatever) but don't change the length of the clip and then
3. put this copy back on an audio track adjacent to the "normal" audio with the clip. You may need to save it first as a different filename. Having got your treated audio back into the timeline on a track adjacent to the original audio, make sure the two are in sync (because you haven't altered the length of the new audio, they will all begin and end at the same point). Then
4. move the start point of the modified audio (don't slide the whole thing along, otherwise you will lose sync) to the point you want the acoustic to change
5. Unlink the video and its "normal" audio track, and end the "normal" audio shortly afterwards (you need to unlink it so the video stays the right length.
6. then mix between the two where they overlap.

That way you will get a smooth transition between your normal audio and the "treated" audio but it will all stay in sync and also you will clearly be able to see by looking at the timeline, where each bit of audio starts and stops. Plus, you will have flexibility as to where to start and end the "treated" audio - you can just play around with the in / out / crossfades of the normal and treated audio tracks.

Most editors (I'm sure premiere pro does) can also do an automated mix, so you could also just keep all the tracks as they were - up to the end of point 3 above - and then mix between the normal and the treated version. Automated mixdown in a video editor is something you need to be careful with as it's easy to lose track of what you're doing and not so easy to see what's going on. I tend to use automated mixing only in the final stages of an edit, and use the technique of splitting and moving audio for specific events during an edit, like the job you're attempting.

What part of that scene's audio do you want to effect changes on? Dialogue, ambiance, foley, music?

@Riker Is that the real sound of the car engine or is the engine sound added? I mean are the radio and engine sounds part of the same audio file? To my ears the "too realistic" sound is the radio and whatever knocking sound comes from inside the car. It kills the effect of racing through the streets and makes you think, oh I'm just inside an ordinary car. You could do this by using a Low Pass filter (LPF) to cut out the higher frequencies since the engine sound is made up of lower frequencies. Some reverb could be added but you will need to adjust the decay time constantly to sort of match the narrow streets, turns etc.

@Riker - It's pretty hard to judge with the low quality clip. I mean, for me, it fits right now. Now if you were to upload a high res version, I would expect way more mastering.

@Riker when it comes to sound design its not about whether things sound "real" or "fake" but if its convincing, that is, believable by the audience. With that said, good film sound is generally not appreciated, because when its good you're not supposed to notice it. The sound should fully compliment the image but not stand out. Strangely enough people tend to believe "fake" constructed sound effects more than real location sound, so that's why tv, hollywood, commercials still does it that way, even though recording technology is better and cheaper than ever.

First time watching your clip, the engine sound was believable to me, but since you asked for a critical ear, I thought about what I noticed which was that little knocking sound, it takes my attention away from the engine, the movement of the car and the mood of those narrow streets. If you keep the original interior sound I would at least try to EQ it, lowering the high frequencies a bit, try making it mono etc.

What will make it work is probably not what you would logically think. Visual images feed the logical part of your brain while sound works on a more emotional level. Ask yourself what is the emotional effect of these shots for the film and how can the sound support it? (Sorry if I sound too smart. I teach sound design so I have to repeat this stuff students all the time).