Weight (on earth) is just the force exerted by Astro when he's standing on the earth. It's given by F=mg at the earth's surface. "g" is a measure of the strength of gravity (gravitational acceleration) at the earth's surface, and "m" is Astro's mass.

In space, assuming he's in "zero-gravity" so you can take g to be 0. Therefore F=m*0=0. So he has zero weight in space.

I'm assuming you want to know how much weight he would have on earth. To get that, you need to figure out his mass (m). The easiest I can think of off the top of my head would be to have him sit on a big spring (or his spaceship's ejector seat!), then launch him from it (basically replace Syphrum's bungee cord with a spring). His energy after launching from the spring depends on his mass (which you want to find) and his speed (which you'll have to measure). This has to equal his energy before launching from the spring (which is easy to calculate). By doing a little math, you can figure out his mass.

Once you know his mass, you just have to multiply it by "g" to figure out his earth weight.

--You could also spin the spaceship around really fast, and he'd "stick" to the wall with some force. If you know how fast it was spinning, and how much force he exerted on the wall, you could calculate his mass that way, too.