Bioshock does not have a manual save option, and the autosave doesn't work for most of us modded console owners. Well, I found the solution all by myself!!(do rep me if it worked for you)

Its all for the time issue we modded xbox owner have. Xbox requires internet connection to maintain proper time(so lame). If you notice when you turn the xbox the date is NOV 22 2005 and the time is always the same too, most probably 5 pm or something. Now here is what you need to do :

1. Go to storage and select the Bioshock Infin, you will see all your saves.
2. Select one by one and look at the right bottom to spot the time/date.
3. Once you have managed to spot it, find the one with the most latest time.
4. This is the trick - If the file with the latest time is for example 7.30pm, just go to your time setting and make the time 7.35pm. THAT'S IT!!!
5. Do it every time before you play the game.