The specific code sample you are describing does not work with partial post-backs, since ClientScript.RegisterStartupScript()
writes JS to the page during the output construction phase of the request lifecycle; whereas a partial postback only updates a selected portion of the page via JavaScript (even though the markup for the entire page, including your startup script, is generated on the server).
To closely mimic what you are describing, you ought to include a Literal control inside your UpdatePanel, and during partial postback set the Text property of the content panel to the script you wish to run:
myLiteral.Text = "<script type=\"JavaScript\">doStuff();</script>";
IMO, a more proper way is to use the client-side API for async postbacks to register an event handler to run when the postback completes:
function endRequestHandler(sender, args) {
doStuff();
}
Sys.WebForms.PageRequestManager.getInstance().add_endRequest(endRequestHandler);
If you need to pass information which was generated during the postback into the handler, you can pass that via hidden fields and grab that from the DOM in your client-side handler.