Hi and welcome to the forum.
So, if your partner wishes to study here in NZ she will need the written offer of a place on a qualifying course at a recognized NZ educational facility. This written offer allows her to apply for the study visa, however she will also have to meet all the other basic criteria like health, character, proof of funds etc etc.
Google study visa for Immigration New Zealand and read all the info available on the INZ website making sure you click all the links available so you don't miss anything.
Assuming she meets all the criteria she would make out a formal application which may be able to be completed online or may only be on paper ?
As the partner of a person holding a study visa it is possible for you to secure a temporary work visa via the partnership category by making your own application. I think its called Partner of a Study Visa Holder Visa ? Once again, Google it and view on the Immigration website for all the details. It is much better to make both applications at the same time, so for example......if you both did it via paper applications you both fill out the necessary individual paperwork pinning all required documents together for each person BUT put it all in one envelope and send to Immigration OR if done separately online you'd have to cross reference each other's application. That will assist the case officer.
Since your application is based on partnership you will have to provide sufficient evidence that the partnership is a real loving stable relationship and that you have been living together for at least 12 months. You will also have to meet all the other usual criteria for health, character etc.
Don't think your ages matter in the above scenario.
If these options don't work out for whatever reason then because of your ages you could both apply for a Working Holiday Visa (WHV). These are open for 18-30 year olds (for the UK). Applications from individuals in the UK can apply for 12 or 23 months and will allow both of you to come to NZ for an extended period and allow no more than a total of 12 months temporary work. It will also allow a period of study.
Better that you also research this visa and it's conditions surrounding study and work and the things you can and cannot do plus all the qualifying info.
Any more questions just ask.
I suppose the route you take all hinges on your partner and the offer of a study place.
Being totally honest you are gonna have to get your skates on. July isn't far away and study visas / temporary work visas can take several weeks to process.