Application for research program is indeed throughout the year. However, it's not just submitting your application and hoping that the said uni will accept you, these are the things you need to do before that, in the case of UM, you need to first find a supervisor
1. Whose research field interests you.
2. Who has vacancy and is willing to accept you.
2. Who has sufficient grant that can support you for the entirety of your masters/phd research project.
4. Talk to him/her about his/her current project(s), or propose one
After that, write a proposal, get your proposal approved by your supervisor/co-supervisors, only then you can submit your application.

Good Luck

Yes by email, there's no telling that they'll reply so to be safe just shortlist a few potential supervisors, send them your CV detailing your research experience and ask them if they're hiring postgrad. The number of postgrad is critical to their promotion, so if they're on the lookout for postgrad, you'll be in luck.

As for finding a part time job, it depends. Some supervisors will ask you to start working in the lab as RA while prepping your proposal and if they didn't you might need to find a part time job because it will take some time for you to write your proposal and getting it approved by your supervisor.

As far as I'm concerned the research in local unis is satisfactory, but sadly the long term prospects are not good as local unis are facing intense budget cuts from the gov each year and this is affecting the number/amount of grant that lecturers can get. The students are feeling the heat too because we're depending on the grants to purchase consumables/reagents for our research project. So, if you're fortunate enough to get an overseas offer, go for it.

This post has been edited by ComposMentis: Jan 31 2017, 10:36 AM