Check your options thoroughly. There's a reason share houses are a thing. Its because apartment shopping in japan is awful. They also exist in single unit style, so you could find one that doesnt make you share the bathroom.
Getting an apartment in japan is a nightmare, if you arent using a realtor. Even tiny leopalace apartments require key money, guarantor, and first month or two's rent in advance. (Around 2800 usd, in my case.)Many places wont rent to foreigners, others wont rent to single women, people your age (whatever that is), etc., etc...they can get really weird with the rules.
A realtor will save you a lot of headaches. Also, on your own it may be impossible to find a place thats renting for less than a year, unless you can find someone who wants a roommate. Leopalace is the only one i know of, but i live in one now, and its seriously overpriced for what you get. You pay for the convenience, lol. But on the upside, its furnished, and comes with internet, unlike basically anywhere else thats not a sharehouse.
The travel expenses are definitely a big thing to consider, but you do get used to Shinjuku, lol. Dont let that put you off, if you find a good place that makes you go through there. A few weeks in and you hardly notice. There are also usually ways around it if you dont mind making a lot of train changes.
I wish I could be more help, but I ended up choosing my job based on which one came with apartment, lol. Trying to get one on my own was too infuriating.