Didn't Reagan support the Brady bill some time later?
Yes, thanks to him there is no right to bear arms, only a government privilege. To buy or sell guns we now need government permission even to our own parents and children according to some states interpretation of this onerous legislation...
Did he sign it though?
Politics, my friend...Brady was his close friend so Reagan felt obligated to speak out on his behalf.
No. The Brady Handgun Violence Prevention Act was signed by Clinton in 1994. Reagan signed the 1986 Firearms Owners Protection Act that modified the Gun Control Act of 1968. While it made somethings easier, like transferring long guns across State lines, and outlawed other things, like tracking ammo purchases, it did hit us with a couple of hidden "poison pills" like the '86 limit on machine gun transfers.
Considering none of the Federal gun laws pass a Constitutional smell test, by anyone other than gun grabbing moonbats, one good SCOTUS case could clear the slate repealing everything back to the onerous National Firearms Act of 1934.