Well it depends on the kind of work you will do. For the most part the system is oriented towards professional work which requires four year degrees. But I imagine there may be some lesser known jobs that don't require it.
You misunderstand it is for the visa, not the job. eikaiwa jobs (English conversation schools) don't usually care about a degree, the visa you need to enter does need the degree. A way around that for some visas is to have enough years professional experience that works in some cases for the engineer visa I heard typically 10 years experience. Eikiawa does not need a degree for the job. I have known people with no degree who were married to a Japanese and did eikaiwa.
I bet your funeral person was married to a Japanese or had been in Japan a long time doign eikaiwa or something and I bet he had a degree for earlier visas. Once you are in for a while if you have enough income you can self sponsor, it is a hard one for me to digest but you are already there. And of course permanent residency is available eventually.
Japanese language school will get you in if you can pay, it is even possible to be allowed to do some part time work while being a student but again the visa ends when your schooling ends and Japanese language school does not give a degree. Now a University in Japan, that could do it, go to school in Japan enter on student visa, graduate university you then have your degree, you are in Japan and easier to get a job get an employer to sponsor.
Marriage also works but be careful with that, many bad marriages this is an important decision be very careful. I as fortunate in mine (I also have a university degree).
There is a visa to come over for cultural activities I am unsure on the limits on that one. Prosletyising is yet another way in, don't know much about their visa.
Post is from US so no working holiday option.
You can get the degree online these days, you have it easy in a way just get the degree and be done with it. You can even do it in Japan. Then you have more visa options if you want to live in Japan. I used to hear jokes for eikaiwa the degree could be in underwater basket weaving it does not matter. I have never worked for an eikaiwa company, I only know some. If you go online just make sure it is accredited, I bet if you have the time you could do it faster as well these days though it might be nice to get a degree in something you could use for what you want to do.
thanks
I didn't think this was a requirement though, I thought I saw that it was just one of the things that you could use to get it but not the only way? But if it's required then that's really bad news...
very doable in fact, I just need to work hard 
