East Cleveland is one the most dangerous, if not the most dangerous part of Cleveland. I would not recommend living there at all if you value your safety. Look into east side suburbs...perhaps Cleveland Hts, which is an older "suburb" area that has more of a city feel, or South Euclid, which is more typical suburban. It's been a while since I lived in Cleveland, but I think these areas should have direct bus routes. Here is the bus website so you can check it out...
Timetables, Maps & Schedules | Greater Cleveland Regional Transit Authority.
I used to place temp nurses at Huron Hospital, and I think you would be ok on day shift. I never took the bus, so can't advise you there, but if you are smart and aware, you should be fine. Stay away on New Year's Eve...the locals are known for shooting off their firearms in celebration.
Good luck with your new job and welcome to America!