To answer your questions first, bluntly, to show you that a bit more work is going to be necessary.
1. Based on the fact that you have (by your own say-so) very little of the required evidence, it's very likely the application would be rejected if sent in on that basis.
2. INZ's own estimates are currently running at 6 - 9 months after allocation to a CO, but this is not a promise, and situations vary enormously.
You need to organize yourselves to get, save, and organize evidence, in order for any application(s) to be successful. You have to PROVE your partnership (which, to INZ, is proving that you are consistently living together), from the beginning (which might coincide with the start of your rental in both names), then show more, varied, evidence preferably at the rate of three things for each passing WEEK, showing that you are at the address, both of you, together, spending time together, and that society around you knows that you are. This is INZ's suggested list
http://www.immigration.govt.nz/migra...artnership.htm, and here are old threads on this forum which will give you more ideas of what can be done.
https://www.google.co.uk/#q=site:enz...tnership+proof The reason this level of proof is thought necessary is that partnership visas, as they only require the partnership to be proved, no qualifications, are often targeted by fraudsters paying citizens and residents to lie to get them a visa. You know you are decent, genuine people, but they don't, and you have the responsibility to put forward your case in a way they can easily check fits their regulations, and verify.
HOWEVER, you have overlooked the existence of the partner-sponsored temporary work visa.
http://www.immigration.govt.nz/migra...milystream.htm As you will see, there is no mention of a qualifying time for this visa, but from other people's experience, we've seen that in practice, INZ take applications with three months'-worth of evidence seriously, and those can be granted a 12-month visa, giving time for the gathering of evidence to continue so that a partner-sponsored residence visa can be applied for when there is 12 months' evidence to hand (including what has already gone in, so the application can be made before the end of the temporary work visa). Incidentally, with all of these, it is usual for INZ to ask for more proof of partnership covering the time of the application just before the visa is issued, to make sure they are not handing it out to someone whose partnership has broken down in the meantime, so don't stop collecting evidence when the application has gone in. Also, keep copies of what was used in the first place, as any later application may not go to the same office so any filed evidence might not be readily available to an official dealing with a later application.
Therefore, it would be possible for your partner to apply for a partner-sponsored temporary work visa at any time when you can see you have really solid evidence of partnership covering at least three months. A good case can be processed very quickly (see your local INZ branch processing times page for current estimates). Whenever a NZ visa is granted, it takes over from whatever one the applicant was previously on, so you will need to look into the fact of whether there might be some impact on your partner's position as a student if he got the partnership work visa before his studies finish (there will be a university department or official dealing with foreign students who will be able to advise), and if so, then time his application carefully. I am thinking, when you have organized your evidence, you might find that you are already quite near to having three months'-worth, or at any rate would be able to get at least that much, well before the close of the academic year, but you don't necessarily, if it would make difficulties, need to use it immediately.
If he is going to apply for the partner-sponsored temporary work visa, given your intention to go travelling this time next year, make sure to check it is issued for multiple entry. They normally are, but occasionally there is a slip. If this should happen, INZ can easily put it right if asked.