This vehicle does not have a serviceable filter. This filter is located inside the transmission case, and it is only replaced when you need to split the case open to rebuild the unit.

So, you will want to follow this guide to service the unit except for the portion of changing the filter.

https://www.2carpros.com/articles/how-to-service-an-automatic-transmission

Please see the info attached on the filter and you will see it is located inside the case.

Thanks

A manual transmission generally doesn't have a filter however you can usually change the fluid in it. Like many of the post 2000 Saturns the Astra is simply a badge engineered Opel product. The manual transmission should take almost 2 quarts of GM 88862472 lube (Basically GL-4 75W/85 synthetic oil) To drain it you remove the bottom plate from the transmission, then you can wipe out the gears and inspect them. To check or add fluid to that transmission you remove the back-up light switch. On a level vehicle the oil should just start to leak out with the switch removed.