Print line after nth occurrence of a match

awk -v n=3 '/<Car>/ && !--n {getline; print; exit}'

Or:

awk '/<Car>/ && ++n == 3 {getline; print; exit}'

To pass the search pattern as a variable:

var='<car>'
PATTERN="$var" awk -v n=3 '
  $0 ~ ENVIRON["PATTERN"] && ++n == 3 {getline; print; exit}'

Here using ENVIRON instead of -v as -v expands backslash-escape sequences and backslashes are often found in regular expressions (so would need to be doubled with -v).

GNU awk 4.2 or above lets you assign variables as strong typed regexps. As long as its POSIX mode is not enabled (for instance via the $POSIXLY_CORRECT environment variable, you can do:

# GNU awk 4.2 or above only, when not in POSIX mode
gawk -v n=3 -v pattern="@/$var/" '
  $0 ~ pattern && ++n == 3 {getline; print; exit}'

Here's a perl one:

perl -ne 'print && exit if $c==3; $c++ if /<Car>/;' file 

With GNU grep, you can also parse its output like:

grep -A 1 -m 3 '<Car>' file | tail -n 1

From man grep:

-A NUM, --after-context=NUM
          Print NUM lines of trailing context after matching lines.  
          Places a line containing a group separator (--) between 
          contiguous  groups  of  matches.          
-m NUM, --max-count=NUM
          Stop reading a file after NUM matching lines.  

With GNU awk you can do:

gawk -v RS='</Car>' 'NR==3 && $0=$2' inputFile