Some companies do help you find an apartment. Either they have company apartments, which would be great because they are great value, or they have an agreement with a rental agency. The benefit of that is that it will be easier for you to rent a place because you donft get turned down for being a foreigner and you have the guaranator covered.
When I looked for a house in Tokyo a few years back in a few rental agency a bit more into the suburbs (somewhere along the Yokohama line) I got literally turned down immediately after entering the rental agency simply for being a foreigner. We finally decided to live in Tokyo (Shimokitazawa) and there it wasnft so tough. However even there the rental agency needed to negotiate with the owner if hefd agree to let his house to foreigners. He agreed and we had a very nice and big house in Higashikitazawa.
The other point about guarantator is that in Japan it is very expensive initially to rent a place. You can expect to pay something like 5 months of rent at once : deposit, 2 months of rent, agency fee, thank you money, guarantator insurance (if you do not have a guarantator).
So if your company is helping you with that , that could be a huge saving. I am not saying you should choose the company based on that, but it is a factor.
An other possible place where to look for an apartment is UR housing. This here is NOT their official page but it is in English and can give you a first impression:http://www.ur-housing.com/
The good thing about UR houses is that you donft need to pay agency fee, key money and donft need a guarantator. They also accept foreigners without problems and itfs on a first-comes first-serves basis.
Their apartments are of good quality and more like for families but you might be able to find something small as well.
I think this is their official page:https://www.ur-net.go.jp/chintai/sp/
Obviously you could also see with Waseda if you could stay as a couple in a dormitory.
As PPs said, companies NORMALLY pay the commuting expenses. If unsure ask them in the interview. Universities donft but if your partner is registered as student shefd get a discount on her commuter pass.
Regarding staying in or outside of Tokyo. The commute leaving the city will be easier (less crowded) than going into the city. Then there are more and less crowded lines and more or less fast train lines. This obviously influences the rental price of your apartment as well.
Enjoy your life in Tokyo!