The mechanisms of updating
bash on your Tiger system are similar for those on later systems. If you're comfortable with compiling and installing your own programs, this may not be outside the realm of possibility, but if you're not familiar with compilers, Makefiles and the gotchas that can come with trying to do it the non-Apple way, you may well be better off leaving it as is. If you're not running Web Sharing, remote login, ssh or other services, you don't have too much about which to worry. CUPS, OS X's printing subsystem, is vulnerable due to the way it interacts with
bash, but the risk here is minimal.
If you really want to examine the steps involved, I wrote this summary based on OS X Lion 10.7.5:
Significant differences will be your starting point. For one thing, you really do want to keep your base
bash --version as the starting point. So, where the Lion and later systems would use v3.2.48 as the base, you'd want v2.05b:
And then you'd download all 9 of the patches from:
Even using my Ask Different article as a template, you may run into problems. Note the comments under the article in which one Snow Leopard user ran into issues because the GNU sources were looking for a different version of
readline in order to compile. You may or may not run into the same problem. You may run into others.
Best of luck with it. As pointed out elsewhere, you'd do well to consider moving from Tiger to a supported version of OS X. Currently, the oldest supported OS X version is Lion 10.7.5. Everything older than that has been end-of-lifed by Apple. Lion will likely go the way of the dodo as soon as Yosemite is released.