Hi rjinjp,
You may want to first start by visiting Australia's Immigration website at immi.gov.au You can review the different visas that you can apply for if you meet the requirements. Yes, there is a visa that has the points system. However, there are other visas available if you are able to be sponsored by a company or a specific region in Australia who have a shortage of a specific skilled worker. There are different lists that you can find in the website to determine if you may qualify for a certain visa.
Some posters from this Australian forum have been able to be sponsored by the company for the job they applied for. It all depends if you have that special skill that they are looking for that company to sponsor you. Two online job websites that I know are mycareer.com.au and seek.com.au
If you are under 30 years old, you can try applying for this if you are a citizen from a country that can apply for this. This will allow you to live and work in Australia up to a year. I have heard of some people getting sponsored by the company that they were able to get a job with. If the company is able to sponsor you, it will allow you to get a different visa to enable you to stay longer in the country to work. In some circumstances, depending on what type of visa you were approved for, you can apply for Australian citizenship if you meet the residency requirements.
I think the only requirement that maybe to be approved besides the participating country is you have to have a certain amount in your bank account. Please review the website about holiday working visa for more information.
I am not familiar about the need of teaching English in Australia without having an English or teaching degree. Since Australia is widely known to be an English speaking country, the demand of teaching English that you do in Japan might not be as high in Australia. Of course, this is just my opinion. However, there could be a particular organization in the country that would be interested in having a specific person to teach English to new immigrants to Australia. I'm not familiar with how to find this information. You might have to do some networking there in Japan who may have friends or relatives that live in Australia if they have any information.
Or the other option is that you can get a student visa to attend a university to take up a course that will give you a degree at the end. This will likely be the expensive route since in order to be approved for this visa, you not only have to be accepted by a university, but you must meet the minimum financial requirement. You can find all the details in immi.gov.au