Is a week is a proper noun?

No, the term 'last week' is a noun phrase, a group of Words based on a noun that functions as a unit in a sentence.

The noun phrase 'last week' is made up of the common noun'week' and the adjective 'last'.

Example uses of a noun phrase:

Last week, we were at the beach. (adverb, modifies the verb 'were')

Last week was really cold. (subject of the sentence)

We caught up on last week's news. (possessive noun; modifies the noun 'news')

We caught up on the news from last week. (object of the preposition 'from')

The noun 'week' is a common noun as a general Word for any week at any point in time.

A proper noun is the name or title of a specific person, place or thing; for example, The Week Publications, Inc. publisher of The Week magazine or Week, a village in Devon UK.