Do they mean the same?
No, they do not mean the same. "Come at problems" means to look for and implement solutions, while "Face problems" means to acccept the difficulties they bring. Perhaps best to think of 'Facing problems" before you 'Come at them'.
Ex. 2) She generally face problems without any difficulty.
I have a couple of problems with your #2:
a. She... faces problems...
b. I don't like the "without difficulty". By definition, problems are difficulty, so there cannot be problems without difficulty. I prefer 'without fear' or some such.