Learn Russian: Ожидание составит минимум часа два. - The wait will be at least about two hours.

Putting the noun before the number is a common Russian trick to mean 'approximately' or 'about'. While два часа means exactly two hours, часа два means about two hours.

Does flipping the word order change which grammar case we use for the noun?

No, the grammar rules remain exactly the same! The number два always requires the noun to be in the genitive singular case. The genitive singular of час is часа, so we still use часа whether it comes before or after the number.

Why does this sentence use составит instead of будет for 'will be'?

The verb составить (here in the future tense as составит) literally means 'to amount to' or 'to constitute'. In Russian, it is the standard and most natural verb to use when announcing totals, prices, or durations, though будет would also be perfectly understood.

Is it normal to use both 'at least' (минимум) and 'about' (часа два) together?

Yes, while 'at least about two hours' might sound slightly contradictory in English, it is very natural in spoken Russian. It simply softens the boundary, implying that the minimum wait time itself is a rough estimate.