Okay, not much we can offer you since you seem convinced that the moving coil (solenoid) is the best and/or only solution to your problem. Your construction is reminiscent of a loudspeaker whose voice-coil (your solenoid) is suspended in a magnetic field (your neodymium magnets) by a flexible "spider" mechanism and whose coil wires are attached (soldered) to braided flexible wires to avoid metal fatigue effects. Perhaps your outer piston cylinder serves a similar "centering" and guiding purpose to that of a loudspeaker spider. In a loudspeaker, the spider also serves the important function of restoring the voice coil to a known position when there is no current in the coil, thus maximizing the back and forth motion possible.
Depending on size, required frequency response, and power handling capability, the linear travel of a loudspeaker voice coil can be quite large, but it is eventually limited by the compliance of the spider mechanism and the edge compliance of the speaker cone. Your design has no such limitation since the solenoid is free to slide any distance within the outer cylinder. You may have a problem with friction between the outer diameter of the solenoid and the inner diameter of the enclosing cylinder.
You may be correct in assuming that your freely sliding solenoid will have less inertia and will respond more quickly than a moving armature located within the solenoid. On the other hand, given sufficient current in the solenoid coil, and a sufficiently strong neodymium magnet sliding within its core, you can probably get equal or better performance from the neodymium magnet (attached to the rear face of the piston) without worrying about how to conduct current reliably to a moving solenoid. That design would also lend itself to a linear bearing to support the piston during its back and forth travel. A stationary solenoid is also much easier to cool since you will likely want to run the piston as fast as possible, meaning maximum current through the solenoid. Note that with either design approach, the piston position and direction of motion is determined by the direction of current through the solenoid. No need for a restoration spring and the mechanical resonances it would introduce. Just reverse the current in the solenoid at each end of the travel of the piston. Of course there will be some "springyness" associated with compressing the air in the cylinder, but presumably you have taken that mechanical consideration into account.
As far as electrically exciting the solenoid is concerned, most folks use a power MOSFET for this purpose. Since you will probably want to reverse the current in the solenoid at the end of the compression stroke, and restore the original current direction on the next compression stroke, a so-called "H"-bridge configuration is normally recommended.
Care must be taken in the design of the "H"-bridge circuit to allow for dissipation of the electrical energy stored in the magnetic field of the solenoid. This must be done to avoid damage to the MOSFETs which are acting as very fast on/off switches. Most MOSFETs have a so-called "body diode" that conducts when the magnetic field of the solenoid load collapses or reverses, thereby protecting the MOSFET from damage but limiting how fast the circuit can cycle. You must also carefully design the gate drive circuits for the MOSFETs to avoid "shoot thru" where two MOSFETs connected between the two power supply rails simultaneously conduct because of gate capacitance effects and poor timing design. There are commercial integrated circuits that implement "H"-bridges with significant current capacity and other features, such as pulse-width modulation (PWM) of the current, with negative-feedback current sensing to control the PWM duty cycle.
About the middle of the last century I had the dubious opportunity to examine a "one of a kind" proportional-flow hydraulic servo valve based on a light-weight aluminum voice-coil actuator, about two inches in diameter, and wound with a single layer of enamel-insulated copper wire. Rare earth (neodymium) magnets hadn't been invented yet, so this puppy used a huge Alnico permanent magnet to create a magnetic field for the voice coil to react against.
IIRC, the voice coil was just the first stage of the valve, controlling flow with a small spool valve attached to the voice coil actuator. That small spool in turn operated a much larger, high flow-rate, spool valve. This "servo valve" was supposed to control a 1,000,000 Lbf (static load) dynamic fatigue testing machine capable of producing dynamic tension and compression loads of ±500,000 Lbf with sixty hertz sinusoidal motion. IIRC, the piston had a diameter of about twelve inches, a stroke of about eighteen inches, and operated with 3000 psig hydraulic pressure. With this much flow and pressure, most engineers would opt for a variable-displacement pump. The contractor opted for a less expensive fixed-displacement pump equpped with a dump valve to modulate the pressure down to 3000 psig.
This California company that won the low bid for the contract to build the system went bankrupt trying to get their valve and actuator piston/cylinder to work to Air Force contract specifications. The actuator piston and cylinder were actually quite competently designed and built, but they decided to save some money by designing their own version of a high-performance hydraulic servo valve. An existing off-the-shelf Moog high-performance hydraulic servo valve eventually got the machine working to the satisfaction of the Air Force... years after the original principals went bankrupt and defaulted on their contract. Let us hope you never place yourself in such a precarious position because you underestimate the complexity of a job.
Good luck with your "flying solenoid" thingamabob.
