What do you need to work in administration at universities? I would like to work in administration at university and I was wondering if there is any specific course/qualification that will increase my possibilities of doing so.

In particular, I was thinking of any job in the "support staff" in charge of general administration. This would include posts related to providing support to students and/or staff with general questions or paperwork but also including organisation of activities, improving processes/how things are done, etc.

I am currently living in the UK so I would also be interested in any specific information for this area.
Usually you apply for these types of bottom- and mid-tier positions through the human resources (HR) departments of the universities. This is the same in the UK or the USA.

If you search on the HR website, they will give jobs and the requirements for those jobs. For example, the first example below is a job posting that I pulled from the University of Illinois HR department's website. You will note that it requires a high-school degree or GED. This is rather typical. When I last hired a staff person, we required an Associates degree or equivalent work experience. The "equivalent work experience" allows us to hire people without an AA degree if we feel that they have enough skills even without the credentials.

The second example below is from Cambridge University in the UK. It notes that you must have at least A levels for this position.

Note that many universities in the USA have union contracts for their clerical and technical staff. This makes applying for jobs as an outsider very difficult because the union contracts usually stipulate that jobs are first filled from job pool candidates (i.e., union members). This may likely be true for the UK as well.