I am having issues at a new job, should I email my boss to let her know what is going on?

IMO, don't bother emailing. Something like this, you should talk to your boss in person. Also, I think it's not a good idea to talk to your boss now since you started your job a month ago. Wait until you pass the probation. Employers are known to put employees on stress test to see how well the employee handles pressure and manages his/her time to get done things. Be patience! wait for a few months then decide.

An office manager can delegate work to anyone under her. If your work is not getting done, then you need to let the manager know when she gives you something else to do that you need to prioritize and where should your work be on that list. Make her aware of your duties and the status.

I wouldn't email the boss, but if there is anything that isn't getting completed and is going to miss a deadline, then you need to make her aware that your going to miss your deadline. When she asks why, then you tell her you are having duties delegated to you that are causing you to get behind. Anytime the manager hands you something to do, ask her where this falls in your list of tasks so that you aren't remiss in your responsibilities.

Don't be a tattletale because that will cause major conflict with the manager and if you tick her off, you may grow to hate coming to work.

Good luck to you.

Face to face with the boss and ask if it is possible to have more time to do both your job and the office manager`s work. Then show what work the o m has given you to do.