The extreme liquid transport properties of carbon nanotubes present new opportunities for surpassing conventional technologies in water filtration and purification. We demonstrate that carbon nanotubes with wettability surface patterns act as nanopumps for the ultrafast transport of picoliter water droplets without requiring externally imposed pressure gradients. Large-scale molecular dynamics simulations evidence unprecedented speeds and accelerations on the order of 1010 g of droplet propulsion caused by interfacial energy gradients. This phenomenon is persistent for nanotubes of varying sizes, stepwise pattern configurations, and initial conditions. We present a scaling law for water transport as a function of wettability gradients through simple models for the droplet dynamic contact angle and friction coefficient. Our results show that patterned nanotubes are energy-efficient nanopumps offering a realistic path toward ultrafast water nanofiltration and precision drug delivery.