Is it okay to ask the professor to let another student continue my Master's thesis topic next year?
In the thesis there is usually a section called "future directions" or "recommendations" in which the candidate can lay out what can be done to improve or scale up the deliveries. Projects being developed in succession by different students are common, at least in my field (biomedical.)

As for whether you will get full credit, the question at its current state cannot really be answered because too many details are missing. For instance, has this likelihood of non-completion been on the advisor's radar? Or are you just dropping the bomb one month before the due date? Who has been causing the delay? Were any barriers or difficulties you faced raised and discussed? How incomplete is the project? All these can go into the consideration of your final grade, and that can of course include a potential "F."

I'd suggest, given the time limit, stop trying to get guessed answers on this forum. Instead, immediately schedule a meeting with your supervisor and make sure both are on the same understanding. Upon discussing with the advisor, come up with a list of tasks that need to be finish to make the Master thesis a reasonably stand-alone work, and complete it. Be realistic and don't be a yes-man (like what you did at the beginning); if the list grows too big, say so and trim it down together. Diligently follow up weekly and deliver the project.

Of course, if the project is grossly behind or substandard, you may want to prepare for an extension or walk away with a fail grade. The results of this discussion will vary depending how professional you have been behaving in this duration and perhaps many other factors. I wish you good luck and feel free to come back and update us by revising your question.