Is 250k~ Yen enough for a 1 month vacation?

Hello,

I'm planning on visiting Japan for the first time in June this year, going solo. I'll have around 250k Yen by the time I go (this is after paying for accommodation). I don't plan on buying many souvenirs. I just plan on eating food, visiting places I'd like to see, and going out with a few friends of mine for drinks and lunches/dinners.

I'll be doing Tokyo - Kyoto - Osaka most likely.

Your main expenses are accommodations, food, and transport. If you exclude accommodations from that figure, you can easily have enough money for a month, provided you don't start throwing money around like a drunken sailor.
It is a huge myth that Japan has to cost you your soul. Living in Japan can be quite expensive, though naturally it depends with where else you're comparing it with. There are huge bargains that foreign travelers can use, including rail/subway passes and special airfares. You can eat quite decently for lunch or dinner for about 1000 yen - or less if you pick fast food, convenience stores, supermarkets, etc. For many, a lot depends on their comfort level - if you insist on having everything "just so" or exactly like back home, it can indeed get very pricey. But if you are flexible and a bit frugal, you can do just fine.

250,000 yen for a month is around 8,000 yen per day given you have accommodation sorted. You can eat very well for less than 4,000 yen/day as food is likely to be your major expense. You can do it for less, but that might get a bit dull and food is one of the nice experiences.
As you are not doing too much travelling around (long distance), transport might not be a big item either. Depending on a few day trips and some planning, within the large cities getting place to place you might get to 1,000 yen/day, but other days would be a lot less than that.
The other main expense you will probably face, unless living like a hermit) are entry fees. A lot of places you might go are free, but others will have entry fees - a lot are around the 500 to 600 yen mark. More expensive places tend to be things like art galleries.

My daily budget as someone who also almost always visits for at least a month at a time:

Food and other essentials - 2,500 yen
Local transport - 1,000 yen
Fun, shopping, everything else - 5,000 yen

Of course, everything that doesn't get used rolls over to the next day.

Multiplied by 30 days this works out to pretty much exactly your budget, 255,000 yen. Personally I think it is fine, but I'll add a few thoughts anyway.

A first-timer may want a little extra to play with than someone like me who has been many times, but you mention that you don't plan on doing much shopping whereas I tend to do a fair bit. In a month I might do two or three somewhat decent shopping trips, spending like 30-40,000 in a day. So you might actually be okay on that front.

Long-distance travel can be expensive if you prefer trains to buses. But on the other hand, I rarely actually spend 1,000 yen a day on local trains. It is probably closer to 500. So again, I have some left over.

I agree with JCT about food. I generally eat out at a decentish restaurant probably two or three times a week. I couldn't do it for 2,500 yen a day. But mostly I eat "normal" food as I always pick accommodation which has proper cooking facilities. I could very easily eat for less than half my budget if I wanted to. Make breakfast for 200 yen, have a 5-700 yen conbini bento for lunch, then make dinner (or eat it somewhere real cheap) for under 500. Overall I think that staying for a longer period (a month), mindblowing food experiences are not necessarily, nor particularly desirable, every single day. Still, I recommend thinking about the food/restaurants you definitely do want to try and budgeting for those plus extra because you are bound to randomly happen upon a few interesting looking restaurants that you might like to try.

Basically I think 250,000 is very doable and you can have a really nice time, but it is still in the territory where you still have to plan a little bit and not just go crazy.