Probable dumb question - you can save at a rest room, return to the castle entrance, get on with normal life and stock up on supplies and then carry on from the rest room you saved at yes?
Keep running out of supplies at the halfway point of the first dungeon!
Yes you can. Basically every new safe room that you reach can be counted as a "checkpoint". Just reach wherever you can, leave the Palace. When you return you can just press R1 and fast travel to the last safe room you left it at.