This work investigates the control problem of discrete-time underactuated mechanical systems with fixed input-delay and matched disturbances. A new control strategy is proposed, which builds upon a discrete-time implementation of the interconnection-and-damping-assignment passivity-based control (IDA-PBC) and extends it in two ways: the disturbances are estimated adaptively; the input-delay is compensated with a recursive algorithm. The resulting control law is constructed from IDA-PBC without solving any additional partial-differential-equation (PDE). Stability conditions are discussed and compared to alternative designs. Numerical simulations for the ball-on-beam system and for the Acrobot system demonstrate the effectiveness of the proposed approach.