Can I walk from Tsim sha Tsui station to East Sha Tsui station to get to Shatin Race course.

Yes, it is very easy to walk from TST to TST East. The signages in the underpass is very clear. You won't get lost. At TST East, take West Rail MTR line to Hung Hom, where transfer to East Rail line and get off at Fo Tan station. Leave from exit F and you can walk to the racecourse.