As Aerovistae suggests, I think confidence in your intuitions, especially as you keep improving in French, is an important factor in the process.

However, I understand that if you weren’t yet sure what rester à quelqu’un meant (if you hadn’t seen it before and didn’t have the English translation), it might be hard to pick exactly the meaning to use among the ones listed.

The two tacks I take in this situation are more or less this:

(A) To understand the phrase as a whole, I will identify a short enough, context-isolated phrase and put that into linguee.fr. In this case I’d choose « il vous reste ». Here’s a match it found:

… je vous indiquerai quand il vous reste une minute.

→ I will give you an indication that you have a minute remaining

With enough of those, I get a sense of which words contribute to the meaning and deduce how I should read such expressions. In this case, a few of these matches would yield « Il vous reste X » → “You have X remaining.”

(B) If I want to carefully understand the phrase piece by piece, I do something like what you seem to have done.

• You accounted for the il using what you know about dummy subjects. The only alternative would be that it refers to some previously mentioned masculine noun. I guess the best candidate here is questionnaire way at the end. It seems like you have a strong enough grounding to know that the first explanation is more likely.
• You accounted for the vous as either an indirect or a direct object.

  You can rule out direct because there is already a direct object: moins de deux mois.

  The indirect object hypothesis is good, and here’s how you can test it. If any of the entries have rester à quelqu’un or rester à quelque chose you know that there is a specific meaning for that formulation. Compare, for example, en vouloir à [qqn].

  Since there is no such entry for rester à on WordReference, you have to fall back to a third explanation for the vous. It could be the à [qqn] of belonging, like « C’est à moi ».

You then combine your hypotheses with the meanings available there. “There – remains – less than two months – for you / belonging to you.”

This more or less makes sense for the piece-by-piece translation; it’s enough to reconstruct the idiomatic saying above, if this is your preferred route.

If it hadn’t made sense, you would need to try combinations of (a) other explanations for pieces like il and vous plus (b) other definitions of the main verb rester below the first.

Again, for the sake of simply understanding a French phrase, this isn’t the fastest route compared to using linguee.fr or even asking here, but it’s one way to do it.

Edit: As lkl writes, since the dummy subject is called the pronom impersonnel in French, some common* verbs that pair with it might appear as a verbe impersonnel, which I’d forgotten about. So if you’re in luck, your dictionary would just have il reste [qqch] as an entry and all you have to figure out is the vous.

_{(* Though it is indeed productive, so you couldn’t exhaustively list all the combinations.)}