Well studying in Japan would be the best option. Even though I would be able to hang on in Japan on my own expense for about 4-5 months, I'm not sure I could do even 2-3 months if I engage into any nihongo lessons.
Don't bother using words like "nihongo" here. Unnecessary.
As for 2-3 months vs. 4-5 months, if you are Bulgarian as your flag avatar suggests, you can stay only 90 days as a tourist.
MOFA: Exemption of Visas [Guide to Japanese Visas]
That's why I thought the best option is to find a job that uses little Japanese like you said, any job, just to be able to fund my life there and learn Japanese to the best I can in the meantime. And when I'm capable enough of doing a job where mainly Japanese is needed... well you know the rest. I'm ambitious and I definitely want to pursue a career.
Forgive me for saying this, but you are perhaps
too ambitious. With your current level of Japanese and only 2-3 months of study (even 4-5 on a student visa), you are not going to be up to a fluency "where mainly Japanese is needed". Intensive study 2-3 times longer than that will be needed at a bare minimum, depending on the exact job you could get.
I've thought about internships. Looked trough some internship in Japan websites, but it was all "paid by me" internships.
I thought you said you could afford 2-5 months on your own. If you are looking for an internship that will pay anything, it will be only for food, maybe also lodging, but nothing more since that constitutes work and a different type of visa. Keep looking for internships. There are plenty.
I have little experience in my field.
All the more reason I say you are unlikely to be competitive here. Get your language skills up and more experience under your belt. Otherwise, best chances you have are as follows IMO:
1. student visa and a lucky break at PT work
2. internship that might lead somewhere
3. intracompany transfer visa, which means you work at that company outside Japan for a year first, then come here to work at their branch (no guarantee on how long it will be, but doubtful that it would last more than 1-3 years)
I will keep trying to find a job, but an idea arises in me. Should I just fly to Japan and try my luck on-site, or not...
I think that if I could find a job there I would be able to learn the language faster than I do it now, becouse I will be exposed to it on a daily basis, and if I don't worry about the funding that would be perfect.
You will probably have to stay in a guesthouse or someplace like LeoPalace21, because most apartments require a guarantor (usually employer). Yeah, you could come here to sniff around, but your credentials right now preclude much luck. My advice is to save your money.