Learn Polish: Skończę ten projekt jutro. - I will finish this project tomorrow.

Because skończyć (to finish) is a perfective verb, which means it describes a completed action. When you use present-tense endings on a perfective verb, it automatically becomes the future tense. You don't need będę; the verb skończę on its own means "I will finish."

Does "skończę" come from the same verb as "kończę" (I am finishing)?

Yes! Kończę is the imperfective version, describing an ongoing action in the present. Adding the prefix s- creates the perfective verb skończyć. Since you cannot be "currently completely finished" in the present, applying the -ę (I) ending makes it the future action skończę (I will finish).

Why doesn't "ten projekt" change its ending since it's the direct object of the verb?

It actually is in the accusative (direct object) case! However, projekt is a masculine inanimate noun (a non-living thing). For these types of nouns, the accusative case looks exactly identical to the dictionary form. Therefore, ten projekt stays exactly the same.

Could I move "jutro" to the beginning of the sentence?

Yes, Polish word order is quite flexible. You could say Jutro skończę ten projekt to put a little more emphasis on the fact that it will happen tomorrow, but the core meaning stays exactly the same.