Is should a noun

No, the Word 'should' is a verb; most often an auxiliary verb.

The verb 'should' used to express obligation or duty; used to express probability or expectation.

Examples:

You should finish your homework before turning on the TV.

We should see the exit sign any minute.

A noun is a Word for a person, a place, or a thing.

The nouns in the example sentences are:

• homework
• TV
• exit sign
• minute