How to replace this NPN transistor with a MOSFET?
Bipolar transistors are cheap and available.  Get some.

This is a pretty crappy circuit in the first place.  It is in no way capable of providing 1 W of AC power into the speaker.  Even ignoring the inevitable off-center biasing of the transistor and any voltage drop accross it, at best this circuit is a 4.5 Vpp AC source with 50 &Omega; output impedance.  4.5 Vpp is 3.2 V RMS.  The highest power transfer happens when the load impedance matches the source impedance, in other words when the speaker is also 50 &Omega;.  That means the load (the speaker) will see half the open circuit RMS voltage, which is 1.6 V.  (1.6 V)2 / 50 &Omega; = 51 mW.  Or about 1/20 of the advertised output power.  With a typical 8 &Omega; speaker, the output power will be considerably lower than that.

Another problem with this circuit is that it puts DC current thru the speaker.  That's generally not a good idea, but at these low power levels won't cause any harm to a normal size speaker.  This circuit is so poor that the additional distortion caused by DC thru the speaker is the least of the problems.

This circuit could work with a N channel MOSFET replacing the darlington, but the same output power calculations still apply.  MOSFETs don't have as well a defined gate voltage threshold where they quickly change between off and on.  Because of that, the bias point produced by the two 100 k&Omega; resistors can't be computed up front.  If this is just for experimenting, use a 100-500 k&Omega; pot for the two bias resistors and adjust it so that the drain sits at about half the supply voltage (4.5 V) when there is no input signal.