What kind of a phrase is 'first class'? It seems to act like an adjective, in the same way that 'up-to-date' does; "a first class seat", "a first class effort".

But it also turns up as an adverbial; "We flew first class". It's maybe a noun, too. I came across this sentence; "The tickets were for first class on the early flight".

Is that last sentence correct, grammatically? Can anyone think of any other phrases, or words, that can be used so flexibly, in terms of grammar?

Isn't it just a case of part of a familiar phrase used in place of the full phrase?

Not in the first example, which is drawn from a rankng system. Then it's just what it appears to be, and adjectival phrase.

But in "We flew first class", it's short for "We flew in first class accommodation." But it's the way language works, isn't it? In casual conversation we're much more interested in cogency and meaning, and much less in good form.

Similarly "The tickets were for first class [seating] on the early flight".

As for correctness, it would depend on the context. In formal writing, they're all incorrect. But in informal speech, they're just fine. It's seldom parsed.