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Hi Bret,
A tutorial for TeamCity is here: https://inedo.com/support/tutorials/buildmaster/teamcity/choosing-specific-artifact-from-teamcity
Future note, the TeamCity tutorial may be moved to https://github.com/inedo/inedox-teamcity/wiki as part of a docs re-organization.
As for controlling Apache, this is done through the httpd command, which can be executed with an Exec operation in your deployment plans, e.g.:
# Deploy to Apache
{
Exec httpd -k stop;
# deploy artifacts...
Exec httpd -k start;
}