Before getting into your specifics, allow me a question. If you want the this pump to output fluid continuously, why did you buy a precise, single shot pump? Just curious.
Trying to create a compatible programmable analog output from and Uno will require additional components as you must first convert the PWM output to a voltage and then either level shift to 10 volts or convert to 4-20ma. The 0-10 volt version is easier but requires use of the 24vdc supply. Creating 4-20ma can be done with Uno's 5 volt supply but it requires a few more components due to the current source.
Ultimately, you can probably connect the 0-10 volt reference input to 10vdc (using a TL431, LM4040 or other means off of the +24 supply), thereby setting to the maximum metering stroke and then controlling the frequency of the square wave going into the START input put to give the desired flow rate.