There are several things about your quote(s) that confuse me.
A: The use of "I" vs "the man".
B: I am unsure about which parts are supposed to be 'still part of the normal flow of the story' and where 'the interruption' begins.
In your 'first version';
I look around the room, wondering where they'll take me. A woman walks
into the room, looks at the man, says "come with me, sir", then
leaves.
Isn't the person referred to with "I" and "the man" the same person? If so, I find it strange to switch to a third-person narrative, unless you have an additional 'narrator' narrating anything that 'the man' doesn't narrate in first-person. If that is the case, however, I think quotation marks are necessary for when 'the man' addresses the audience, but I get the sense that this isn't what you want. It also seems that way from your 'second version';
I look around the room, wondering where they'll take me. A woman walks
into the room, looks at me and says, "come with me, sir", then leaves.
I guess I'll follow her.
Still, when it comes to which parts belong to the normal flow and which to the interruption, neither of the versions 'read easily' to me. I would suggest you come up with a clear way of showing when the narrator is addressing 'the audience' and when he isn't, because in the second version, the two sentences seem to be directed at the same audience/be part of the same 'category' - as if the woman entering is still part of the story that the man is narrating.
Suggestions
I see at least three ways you can make it more clear what is what:
1: Switching the tense - 2: Indentation / different formatting - 3: Quotations for the 'real-time events'.
1: Switching the tense:
@Grace gives one example of how you could do this in another answer. If you prefer the present tense for the main part of the story, perhaps you can switch it to the opposite and have the interruption be in the past tense. It may be strange, but it may work if you make it clear what is happening.
2: Indentation / different formatting: Perhaps using indentation or different formatting for the interruptions could make it clear that they're different from the main narration. For the sake of demonstration, I will assume that 'I look around the room, wondering where they'll take me' is part of the main narration. Examples;
V.1: Indentation
V.2: Switching sides

3: Quotations for the 'real-time events': Alternatively, you could put anything 'happening in real-time' (as in, interruptions) in quotations that show that the narrator is not addressing the audience - this could require some re-writing depending on what style you end up preferring, but it seems that's likely the case anyway. These real-time interruptions could be mixed with the formatting examples above for extra clarity. Example;

You could then add to the real-time dialogue if you want to make it even more clear what is happening. You could even write longer parts in that format if you want to explore "that part" of your story.