Learn Hardcore Hebrew: חשוב שתבדקי אם יש רוכסן טוב בכיס הפנימי של התיק. - It is important that you check whether there is a good zipper in the inner pocket of the bag.

תבדקי = you will check / you should check (addressing feminine singular)
If you were speaking to a man, it would be שתבדוק
If you were speaking to more than one person, the form would change again

Hebrew verbs often show the gender and number of the person being addressed, unlike English.

Is תבדקי really future tense? Why does it mean something like check or should check here?

Yes, תבדקי is formally a future-tense form, but after expressions like חשוב ש־ it often has a meaning closer to:

So:

Together, חשוב שתבדקי means it’s important that you check.

This is very common in Hebrew: future forms are often used after ש־ to express something like a recommendation, expectation, or necessity.

Why does the sentence start with חשוב and not זה חשוב?