You can insert arbitrary PowerShell script code in a double-quoted string by using a subexpression, for example, $() like so:
"C:\temp\mybackup $(get-date -f yyyy-MM-dd).zip"
And if you are getting the path from somewhere else - already as a string:
$dirName = [io.path]::GetDirectoryName($path)
$filename = [io.path]::GetFileNameWithoutExtension($path)
$ext = [io.path]::GetExtension($path)
$newPath = "$dirName\$filename $(get-date -f yyyy-MM-dd)$ext"
And if the path happens to be coming from the output of Get-ChildItem:
Get-ChildItem *.zip | Foreach {
"$($_.DirectoryName)\$($_.BaseName) $(get-date -f yyyy-MM-dd)$($_.extension)"}