Cast varchar into int, converting any value under 0 to 0
As TINYINT only allows 0-255 you could also use 

ISNULL(TRY_CAST(YourCol AS TINYINT),0)


The TRY_CAST above returns NULL if supplied an invalid input for the datatype (including -ve numbers) and I would be minded to leave it that way.

But if you must change these to 0 you can wrap it in an ISNULL