Can I use a 1KW inverter along with a motor starter to power a water pump?
The link is just a standard 4 pole relay it will not help other being an on off device. A motor inverter is a device that takes line voltage and adjust the voltage and frequency.
With a very old motors it is not advisable to use inverters or soft starts because of the harmonics generated will cause the bearings to be destroyed in a matter of months. Pump motors don't do well with soft starts because they slowly ramp the frequency or pulse full power then pause pulse pause until at full speed then a internal contactor locks the line to the load. With each delay in pulses the pump slows down and causes more heating.
Bearing failure was a huge problem in the early days of VFD’s (variable frequency drives). E series high efficency motors solves that problem. Here is a powerflex 40 
From e-bay so you understand what a motor inverter or VFD is. With a VFD a relay or contactor is not needed. The Bestek 1KW may not be able to run the motor as these types of inverters usually will not push a pure inductive load like a motor.