I would stitch this motif on the fabric of my choice and attach it to the apron with Velcro. This way it can be laundered separately. You are bound to get grease, sauce and who knows what on it and constant washing and bleaching would most likely damage the motif.
Williams-Sonoma sells really nice aprons for both men and women and you can have initials put on. They also hold up well under repeated washing and bleaching.