Learn Romanian: Premiera spectacolului este diseară. - The premiere of the show is tonight.

This is the genitive case, which shows possession (translating to "of the show"). Spectacol is a neuter noun, and in the singular, neuter nouns take the exact same articulated genitive ending as masculine nouns: -ului.

Could I use la noapte instead of diseară to say "tonight"?

Not in this context. Diseară translates closer to "this evening," which is when a theater or movie premiere naturally happens. La noapte specifically refers to the late hours of the night when people are usually asleep.

Why is spectacolului placed after premiera? Can I reverse the order like in the English "the show's premiere"?

In Romanian, the noun in the genitive case (the possessor) almost always directly follows the noun it possesses. You cannot put it before the main noun, so saying spectacolului premiera would sound grammatically incorrect.

Why does the word premieră end in -a here instead of its dictionary form -ă?

Because it has the definite article attached. For feminine nouns ending in -ă, you replace that letter with -a to say "the". It must be articulated here to grammatically support the genitive noun spectacolului that immediately follows it.