One way:
Sample file
$ cat file
Cygwin
Unix
Linux
Solaris
AIX
Using sed, replacing the next line after the pattern ‘Unix’ with ‘hi’:
$ sed '/Unix/{n;s/.*/hi/}' file
Cygwin
Unix
hi
Solaris
AIX
For your specific question:
$ sed '/<key>ConnectionString<\/key>/{n;s/<string>.*<\/string>/<string>NEW STRING<\/string>/}' your_file
<key>ConnectionString</key>
<string>NEW STRING</string>