My last Japan trip was 4 or 5 years back and spent time in multiple big cities with an express train pass. I think I budgeted a grand for the flight, a grand in food and hotel and spending a week. But with inflation being what it is I'd want to rerun the numbers based off of what flights and hotel/hostels I could find and assume 1k for just food and fun per week. I think there are active data sheets online that talk about the average cost of eating out in Japan right now.

You want to visit for "a few weeks" so I'd say plan for 2k + flight + hotel/hostel + train tickets/pass. I'd bet you spend less than 4k total for that time.

I like to visit 1 major city every 4-7 days, I normally do travel in, 5 days, travel out. So two weeks would let me see 2 major cities and a couple day trips or 3 shorter stays at 3 major places. Some cities are cheaper than others which is something to consider and how you eat out also dictates your budget more than anything. You could eat in Tokyo for dollars a day at gas stations or you could splurge on sashimi every night and find yourself burning money by the fist full.

I'm a big foodie so that's where the 1k per week comes from.