it is important that you have terminated and actually stopped working in your part-time job BEFORE you apply for dagpenge as a graduate. Because otherwise you will be regarded as self-inflicted unemployed, and get a quarantine for 3 weeks, where you will not receive dagpenge.
It is not a problem to sign up for A-kasse and get graduate status registered in A-kasse, even if you have job. The registration in A-kasse/graduate rights will not be affected by the fact that you work on the day you get registered in A-kasse. It is only relevant whether you work or not, on the day you apply for dagpenge - in that case you will be in risk to get the quarantine of 3 weeks, and you can off course not receive dagpenge for the days you work/have income.
It is very important that you get registered with graduate-status at latest 14 days after graduation, regardless of you have stopped working or not. This will give you the right to begin to receive at any point during the next two years. As a graduate you do not need to meet other requirements.
You must decide with your self what you want to do about your part-time job, but as mentioned you will probably get the 3 week quarantine (voluntary/self-inflicted)...so its depend on a calculation I think, to decide what is best in your case.
Another option could be to keep the part-time job and get supplementary benefits. We will recommend you to talk with the A-kasse about this after you have signed up, in case it is an interesting option for you.
Anders Weber, www.a-kasser.dk