Sure, you can run a pipe under a walkway. The trick is to dig a pit about 4' wide (the perpendicular 'direction' of the walkway) and 3' wide (in a parallel direction to the walkway.) Then, dig your pipe about 18" down, and under, your walkway. Do it by hand and use iron pipe for your plumbing. ONLY dig the burrow to the diameter of the piping, no more. That way, you minimize any potential of the dirt collapsing under the walkway. I've done it by hand and from both sides of a walkway. I use any type of 'proding rod' I can find to be sure I'm 'on line'.
YOu want a minimum of 1/4" per foot as a minimum slope for water to 'run'. BUT, have you considered a dry well instead??? It may be the better 'answer'. My best to ya and hope this helps.