Synchronous speeds are a function of the applied frequency and the number of poles, governed by the equation

120 * (frequency in hertz) / (poles) = (speed in rpm).

Adjust the ratio of frequency to poles to achieve the desired speed.
(example: a 4-pole design would require a line frequency of 333.33 Hz … which means operating on either an adjustable speed drive or on a dedicated high-frequency power system.)

Once you’ve got the electro-magnetics sorted out, it’s a matter of manufacturing to the mechanical constraints associated with the rotational speed.

Well, depending on the power rating, and on the required reliability, I believe it’s very simply. The biggest problem would be to get a variable frequency drive, or other power supply to provide a 3-phase output frequency of about 500Hz.
An automotive alternator should be able to operate relatively reliably at your required speed, and it can probably deliver around 1.5kW at that speed.
In order to make it permanently magnetised, we just have to disassemble the rotor, take the rotor windings out, and replace them with some ring-shaped permanent magnet. We may possibly also use a number of individual smaller permanent magnets embedded in some non-magnetic material such a copper or aluminium between the two half-shelves of the rotor. Depending on the construction of the alternator, we may need some machine shop to pull the rotor halves apart, perhaps machine some material away to make room for the permanent magnet(s), and to press the assembly back together again, and to balance it afterwards.