Hi.. Welcome to Singapore! 
Taxi Drivers usually dont have such a practice. But you can
try..
The meter fare from Changi Airport to Woodlands is about $16. Add
that to surcharges of $5, + flagdown $2.40, thats already
$24.
On the other hand, i heard of a scheme, which travellers from Johor
to any location in singapore, pay S$30 per taxi ride. So from
Changi, this may be $40 instead due to surcharges
However Singapore Taxis will NOT bring you to
Senai Airport. Instead, they only bring you to a place in Johor
Bahru Town Centre, aka "Kota Raya". At Kota Raya, Malaysian Taxis
will bring you to Senai Airport (30-50mins ride) for RM30 (about
S$14.00).
So add these 2 up, total fare may be S$40 + S$14 = S$54.
So, in other words, theres really no direct method to Senai from
Changi on Public Trasnport. Minimum is to change taxis at JB Kota
Raya.
hope this helps