One quick and hacky way to do it is to advance along the curve using very tiny increments of the curve parameter, s, accumulating distance as you go until you reach the distance you want to move. It's not as elegant as a numerical solution, but it's easy to do and should probably be fast enough for most applications.

You could do what FLeBlanc suggested. Calculate position for some value t and than find the distance between current position and last position. Check to see if distance is greater or smaller than the distance you wanted to pass. If it is then move along that distance by the amount you want.

Yeah, as far as I know, there's not a simple arc-length formula you can plug a parameter into. You'd have to do some numerical integration at some point. I guess FLeBlanc's method is a type of numerical integration as well, but it's way more suited to a gaming application.