It really depends on the company.
Typically, the screening process will be done only online, often with some coding assignment plus video chat interviews. I’ve heard of companies flying out people for interviews though.
Working visa processing time is basically impossible to predict, as once it is submitted to immigration, you just need to wait. Typically it is two to three months, but I’ve heard it taking as much as six months.
Relocation packages depends on the employer. If it is a big established company, their probably going to be paying for everything related to your move. For smaller startups, typically they’ll just pay for your flight. Some won’t even do that though. No matter what, someone from the company should help you get established in Japan with stuff like renting an apartment, contracting a mobile phone, etc.
With working visas, there’s two main ones.
The Engineering visa, which companies normally use, and just requires a relevant university degree or ten years working experience. This visa isn’t tied to your employer, so you can change without issue.
The newer option is the Highly Skilled Professional visa, which is on a points system, and basically requires a relatively high salary plus university degree plus points for some other things. The visa offers a bunch of “perks”, including a faster track to permanent residency. This visa is tied to your employer. However, if you meet the requirements for this visa, you’re almost guaranteed to be able to get an Engineering visa as well, and so changing jobs shouldn’t be an issue.
Your main initial expense is probably going to be rent. Move in cost can range from one month’s rent to six month’s rent upfront. There’s other articles explaining more about this if you search around. If you’re single, the easiest option to get started is shared accomidation, as it typically has no or minimal upfront fees, and is usually furnished, so you can get settled in before figuring out where you want to live longer term. This costs probably ¥100,000 per month if you’re living someone central. If you go with something like that, ¥500,000 to ¥1,000,000 is probably a safe reserve of cash to have.