I don't know how fast astronauts fall, but in the interest of intergalactic tube science, I'll just tell you everything I do know.
The first thing I did on mushroom fleet was blow up my first ship. I respawned in 0,0,0 with no cores. The closest shop is the spawn shop in 2,2,2. I am of the opinion that when I do ridiculously stupid things, that I should solve my own stupid problems - so I pointed myself at the spawn shop, in astronaut mode, and started "walking". I timed roughly how long it took me to travel .1km (100m) using the waypoint marker and got 20 seconds - about 2m/s - and then divided that by how far I had to go, which is how I found out that I am an idiot.
Mushroom fleet has double speed limit (150m/s) and substantially larger sectors (8km?). I've walked around considerably in astronaut mode on both mushroomfleet and singleplayer (where my settings are default), and never noticed any substantial difference in speed. This leads me to believe that server speedlimit doesn't affect astronaut speed. I don't have access to my starmade installation right now, but you may be able to find something in server.cfg that determines astronaut speed.
Fall speed is slightly different (and obviously faster) than walk speed, but I would bet they're related. Intuitively, I suspect that you either immediately reach top speed (if you are falling, you are always falling x m/s), or if you do accelerate, you reach top speed quickly. I would put my money on an easy number (like 5x walk speed).
For real science, there's this:
Fair warning that "apoplectic" is supposed to be "epileptic". They fall through twice, from :09 to 1:16, and from 2:12 to 3:19. That's 1:06 +- 1s, which is pretty good data - or would be, if we knew how big that thing was. It was built by
BinHawkins2013, if he knows how big it was offhand.
Alternatively, you could build your own tube of known length (crystal spiral optional) and time yourself falling through it. I think you need at least 30fps to not lag, but I'm not sure about that.
I should also point out that no intergalactic tube system will beat out a decent jumpship (sorry). 10 second charges are fairly easy to get. At 8 sectors/jump, with 2km sectors, that's 1.6km/s. To be perfectly honest though, if you actually built an intergalactic tube network, I'd probably use it at least once.