Maybe something like this?
$ cat foo.input
foo
foo
<key>some key</key>
<date>some date</date>
bar
bar
<key>SULastCheckTime</key>
<date>2013-08-10T00:27:40Z</date>
quux
quux
$ awk '/<key>SULastCheckTime<\/key>/ { toggle=1 } toggle && /<date>.*<\/date>/ { gsub(/<[^>]*>/, "", $1); print; exit }' foo.input
2013-08-10T00:27:40Z