to be entitled to unemployment benefits there is among more these two requirements:
1. You must have been member of an a-kasse for at least the last year
2. You must have had normal paid work for at least 1.924 hours during the last 3 years.
Be aware that ONLY working hours during membership-periods is counted.
So if you only have been member of an a-kasse the last year also the 1.924 hours has to be counted the last year.
But if you have been member (before but within 3 years) you can also count these hours in the total of 1.924. Membership-periods from different a-kasser can be used when counting working-hours.
Hope it sounds resonable for you.
As I read your question you have never been member of an a-kasse, so if you only have 9 months left now of your contract it will be difficult I think for you to get the 1.924 working hours. If thats the case then you can not get unemployment benefits even not after one year membership.
But we still will give the advice to join an A-kasse now and then hope you can get the "last 3 months of work" after your current contract ends. Any kind of job could be fine to meet this requirement.
You do not write what country you are from. If you are from an EEU-country and if you have beeen member of an A-kasse in your own country then you should be aware that it is possible to transfer membership-periods and workinghours. But you must ask your a-kasse about this, because it is very specific rules.
You can read more here: http://a-kasser.dk/benefits.html