Metro to Newcastle, Go North-east X9/X10 to Middlesbrough, Arriva X93 or X4 to Whitby. All would be covered by a £10.50 North East Explorer day ticket.
13.04 Metro to Haymarket might just allow you to get the 13.30 X9 from Eldon Square, arr Middlesbrough 14.55 then 15.20 X93 arr Whitby 16.33.
12.52 Metro would be better for the connection, if you missed the 13.20 then it would be an hour later into Whitby by X93 or a shorter wait in Middlesbrough for the slower X4 arriving at 17.25