The RT installer will not configure Apache for you (or any other web server, for that matter).
I would start by checking that you have enabled the FastCGI module:
sudo a2enmod fcgid
I would also update a few settings in /etc/apache2/mods-available/fcgid.conf to better handle timeout and large attachments:
<IfModule mod_fcgid.c>
FcgidConnectTimeout 20
FcgidIOTimeout 600
FcgidMaxRequestLen 1073741824
<IfModule mod_mime.c>
AddHandler fcgid-script .fcgi
</IfModule>
</IfModule>
Then I would make sure your site definition includes the specifics for RT - example below for Apache 2.4.
<Location "/rt">
Require all granted
Options +ExecCGI
AddHandler fcgid-script fcgi
</Location>
Official documentation here.