Learn Hardcore Russian: Вам стоит подождать минуту: сейчас администратор найдёт ваш билет. - You should wait a minute: the receptionist will find your ticket now.

Вам is the dative case of вы (to you). In Russian, advice/necessity is often expressed with an impersonal construction like кому (dat.) стоит/нужно/можно + infinitive.
So Вам стоит подождать literally works like It is worth for you to wait → You should wait (polite/formal).

What exactly does стоит + infinitive mean here? Is it about “costing”?

стоить can mean to cost, but in the pattern стоит + infinitive it means it’s worth / you should / it would be a good idea to.
It’s softer than a command and sounds like polite advice or a recommendation.

Could I replace Вам стоит with a more direct Подождите?

Yes. Подождите минуту. is a straightforward polite imperative: Please wait a minute.
Compared to that, Вам стоит подождать минуту feels a bit more like advice/explanation (often used by staff: You’d better wait a minute…).

Why is it подождать (perfective) and not ждать?

подождать is perfective and means to wait for a short while / wait a bit (and then stop waiting)—a bounded action. That matches минуту (a limited duration).
ждать (imperfective) is more like to be waiting / to wait in general and can sound more open-ended unless you add context.

Why is минуту in the accusative? Shouldn’t it be “for a minute” with a preposition?

Russian often expresses duration with the bare accusative (no preposition): подождать минуту / час / два дня = wait (for) a minute / an hour / two days.
So минуту is accusative singular, showing how long the waiting lasts.

Does минуту mean exactly one minute, or just “a moment”?

Often it’s approximate—like English a minute meaning a moment / a bit. It can be literal, but in service situations it frequently just means shortly.

What’s the function of the colon (:) in this sentence?

The colon introduces an explanation or reason:
Вам стоит подождать минуту: сейчас администратор найдёт ваш билет.
= You should wait a minute: the administrator will find your ticket now/shortly.
A dash could also work in some styles, but the colon is common for reason/explanation.

What does сейчас mean here—“right now” or “soon”?

сейчас can mean both right now and in a moment/soon, depending on context.
In this kind of sentence, it often implies very soon / any second now: The administrator will find it in a moment.

Why is it найдёт and not находит?

найдёт is future perfective of найти and means will find (successfully, as a completed result).
находит is present imperfective and means finds (habitually/regularly), not a one-time future completion. For “is finding (right now)” you’d usually rephrase: сейчас ищет и найдёт (is looking and will find).

What case are администратор and ваш билет, and how do I know?

администратор is nominative (the subject: the administrator).
ваш билет is accusative (the direct object: your ticket). You can tell because найдёт (will find) takes a direct object: найти что? билет.

Why is it ваш билет and not вашего билета?

With найти you normally use the accusative: найти что? билет → ваш билет.
вашего билета (genitive) would be used with negation (не нашёл вашего билета) or in some “absence/lack” contexts, not in a straightforward positive “will find” statement.

Is this sentence formal/polite, and can I say it to a friend?

It’s polite and typical of customer-service speech. You can say it to a friend, but it may sound a bit “official.” With friends you’d more likely say something simpler like Подожди минуту — сейчас найду (Wait a minute—I’ll find it now).

How is this sentence typically stressed/pronounced?

Common stresses: Вам стОит подождАть минУту: сейчАс администрАтор найдЁт ваш билЕт.
If you want a rough transliteration: Vam stoit podozhdat’ minutu: seychas administrator naydyot vash bilyet.