I don't think you'll lose much fluid changing the filter, but the screen is a different story. Is there a reason you think the screen needs cleaning? Scheduled maintenance or some other reason? If you're not having any problem (like jerky or sluggish hydraulics), you might think twice about cleaning the screen unless the maintenance schedule calls for it.
If I need to clean the screen but not replace the hydraulic fluid (like when my backhoe cylinder had seals breaking down) here's what I do. I run my tractor until it's hot (I actually use the tractor so the hydraulics heat up). Then I clean off the grime around the drain plug, and then drain the fluid into a clean bucket/tub (be sure it's big enough to hold all the fluid). Then I put that same fluid back in. If I had any reason to think it was contaminated I'd run it through an old tee-shirt as a filter, but I've never done that with hydraulic fluid. That fluid must still pass through the screen and filter before it gets to the pump, so it can handle minor contamination. Of course, you should replace the fluid with new fluid according to your maintenance schedule.
Be sure to examine your screen for bits of rubber which could indicate that the seals in one or more of your hydraulic cylinders are deteriorating.