You should ask yourself, do you want the Manager job or not

First of all, you should ask yourself, do you want your manager's job or not.  If you want the job, then apply for it. If you don't want the job, then don't apply for it. You should understand higher position job not only means higher salary, it also means more responsibility.

If you decide to not apply for the job. Then you should tell your boss, and let him/her know you are not interested in the position, but you are willing to cover the position until they found a new person; however, you are expecting a Acting Manager salary at the mean time due to the kind/amount of work you are doing.

Once the new manager is settle down, then you can release the acting manager position and salary, then go back to your normal salary.