[susan53]

You're mixing up word class and sentence function here. Both these phrases are noun phrases consisting of a pre-modifying adjective (last) and noun (night/spring).

But their function in the two sentences differs. In
Did you see the game on TV last night?
last night has an adverbial function ie it modifies the information in the clause by limiting it to a certain time. It's therefore a time adverbial.

But in
Last spring was a joyous time
last spring functions as the subject of the sentence

I have to say there are other complications which I'm not going to go into because I don't fully understand the logic behind them. But basically :
- think of nouns, adjectives, verbs, adverbs etc as word classes
- think of subject, object, adverbial etc as sentence functions.