I spent 13 years designing electronics of this exact nature: three phase induction motor reduced voltage soft starters and variable frequency AC drives. I spent the last few of those years as a VFD applications engineer helping customers select and configure this equipment for various loads and industries around the country as well.
You will not be able to build something that is cheap and safe. The voltages and currents involved are well beyond the safety margin of a hobbyiest, especially someone who is openly avoiding buying commercial units in order to save money. Don't do it!
While the theory behind AC motor control is very straightforward, the detail level work (heat sink sizing, snubbering, gate drive requirements, de-sat protection, motor overload calculations, bus capacitor protection, etc.) can be quite tricky to get down, especially with heavy duty cycling and regenerative power modes which a carnival ride will CERTAINLY be generating. I strongly caution you against trying to build something of this nature unless you have significant experience not only in microcontrollers and embedded systems design but also significant experience in power electronics and three phase circuitry. People get hurt and killed building this stuff.
My first question for you is whether speed control is really required, or if you only require a soft start up and slow down. Do you vary the speed of the motor once it is started? If not, you may be able to get away with a MUCH cheaper reduced voltage soft starter. These units act like three phase light dimmers; they only adjust the applied voltage to the motor. You will not have a lot of torque at low speeds, but with the right design of motor (NEMA class D) you can achieve exactly what you're after with a fraction of the cost and maintenance.
If you really do need to vary the full-load speed of the motor then you are more or less stuck using a variable frequency drive. As you are aware these are expensive and if you buy cheap you are likely to replace them sooner due to your high surge current (they call this "constant torque") application. What I would definitely recommend doing if this is the case would be to contact various manufacturers (Allen-Bradley, Cutler-Hammer, SAF drives, Benshaw, Yaskawa, etc.) and ask for reconditioned units. Ask for a drive capable of delivering 150% rated current for 30s (this is usually known as heavy duty) or size the drive 30-50% larger than your nominal current rating. You will also likely be running off of generator power which is notorious for being undersized and prone to brownouts and surges as the load requirements change with the state of the equipment being run. Drives don't like that (voltage sags cause current spikes as the motor starts slipping and surges can cause you to overvoltage the bus capacitors) and have a tendency to either fault out or blow up.
I am all about the little guy building something and saving a buck, but this is not the type of project to do this on. If you really want to build a three phase AC drive, start with a little 10HP 480V motor with a hand brake on a test bench. You have all the potential for experiencing the pants-filling sensation of an H-bridge failure or a bus capacitor explosion two feet from your head but without the potential lawsuits and loss of life (except perhaps your own).
For safety, mechanically disconnect the motor from the printing press.
Some printing equipment can pretty much self destruct if run backwards.
AC/DC motors have brushes. Speed is controlled by voltage and physical load.
Generally two wiring schemes are used, parallel, and series.
Series wired, have good starting torque, and poor speed control,
so should not be started without a physical load.
Common series examples; car starters, vacuum cleaners, hand power tools.
Parallel wired, have better speed control, and lower starting torque.
Investigate the motor wiring. An ohm meter reading of each pair of wires will help.
Many AC/DC motors have access to the brushes on the outside of the motor.
If this is the case, carefully remove one of the brushes (paying attention to how it is installed).
Then re-test the wiring, to identify which pair power the armature through the brushes.
The press manufacture may be able to help, if they originally installed the motor.
If all else fails, plan on parallel wiring for the motor.
Many armatures have very low resistance, so would draw excessive power on starting.
The inductor may have been wired in series with the armature to reduce the starting surge.
Some printing equipment starts at a lower speed, then comes up to running speed.
The variac may have had an automatic power reduction mechanism when powered off,
so when started, it would start slower (lighter load), then come up to speed.
If this is a small press, the operator may just set the speed and have a start/run switch.
The motor wiring also determines motor turning direction.
Verify motor direction, before mechanically reconnecting.
Once connected, turnover the press by hand first, to see nothing is binding before power is applied.
Best Answer
No.
A Variac reduces the voltage without modifying the sinusoidal waveform. A Variac is a variable-ratio autotransformer.
The AC motor controls that provide good motor performance are variable frequency drives (VFD). They simultaneously reduce the voltage and frequency with the voltage generally proportional to frequency. It would be quite difficult to make a VFD reduce the voltage without reducing the frequency. A VFD rectifies the incoming voltage to provide DC. It then uses switching techniques to synthesize an approximation of a sine wave. With an AC motor load, the current is quite sinusoidal. With other loads, results will vary. Since three-phase motors are best for this type of control, nearly all of the many models and brands on the market have three-phase output.
The "motor controller" for which a link is provided in a comment, is almost certainly a triac voltage control. It is more suitable for dimming lights and controlling heaters than it is for controlling motor speed. It would work for controlling a universal motor and to a very limited extent, some types of single-phase induction motors driving a fan or centrifugal pump. The output waveform is shown below for three voltage levels: (A) output voltage nearly equal to input voltage, (B) output slightly lower than input voltage and (C) output voltage much lower than input.
Source: https://commons.wikimedia.org/wiki/File:Phase_control.svg
Alternatives
You will not likely find anything comparable at a reasonable price. You should probably consider more carefully what you really need. You may be able to find an assortment of fixed-ratio transformers that will satisfy some of your needs. For low power, you may be able to use a variable transformer for a toy train. For a higher voltage at an even lower current, you can use a fixed-voltage transformer to step up the output.