Learn French: Excusez ce petit problème, merci. - Excuse this small problem, thank you.

This is a polite command or request. In French, you give a command by using the verb form for tu, nous, or vous but leaving out the pronoun. Here, it uses the vous form to be polite or to address multiple people.

Why do we use ce for "this" instead of cette?

Ce is the masculine singular word for "this" or "that". It must match the noun it points to, which is problème. Even though problème ends in an "e" (which often indicates a feminine word), it is actually a masculine noun! If it were feminine, you would use cette.

Doesn't the adjective usually come after the noun in French? Why is it petit problème?

Most adjectives do come after the noun! However, a small group of common, short adjectives go before the noun. A helpful trick to remember them is the acronym BANGS: Beauty, Age, Number, Goodness, and Size. Because petit describes Size, it goes before the noun.

Do I pronounce the final "t" in petit in this sentence?

No, the final "t" is silent here. You would only pronounce it if the next word started with a vowel sound (making a link called a liaison) or if you were using the feminine form, petite. Since problème starts with a consonant, the "t" stays silent.