Мне осталось чуть-чуть времени до встречи.

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Breakdown of Мне осталось чуть-чуть времени до встречи.

• время: the time
• до: before
• остаться: to remain

Questions & Answers about Мне осталось чуть-чуть времени до встречи.

Why is the pronoun in the dative: why мне and not я/меня?

Russian uses the dative to mark the experiencer/possessor in impersonal “have/left” statements. Мне осталось... literally means “to me remained...”. Same pattern as мне холодно, мне 20 лет.

Can I say У меня осталось... instead of Мне осталось...?

Yes. У меня осталось чуть-чуть времени до встречи is very natural and perhaps more straightforwardly possessive. Мне осталось... is also correct and common, especially in impersonal constructions (and with infinitives: мне осталось позвонить).

Why is осталось in neuter past? Shouldn’t it be present?

This is a common impersonal perfective: the neuter past expresses a present result (“there remains/has remained”). You can use the present imperfective too: остаётся. Nuance: осталось = result (“left now”), остаётся = ongoing process.

So can I say Мне остаётся чуть-чуть времени до встречи?

Yes, it’s correct. It sounds a bit more like “there is (only) a little time remaining” as a current process. In everyday speech осталось is very idiomatic here.

Why is времени in the genitive?