Remember how in my first reply, removing the ^ eliminated the requirement that the match occur at the beginning of the string? Guess what. Putting it back will reinstate that requirement:

my @arr = grep /^"/, @arr_s;
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One metacharacter at a time, one question at a time is a really inefficient way to learn about regular expressions. Please have a look at perlretut, perlrequick, and perlre. The first two of those documents will take a half hour to 45 minutes to get through, and then you'll be ready to save the day with regular expressions. ;)

Update: I can see I missed the part of your question that says words shouldn't end with ":

/^"[^"]$/
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That would accept anything that starts with a ", but contains no " characters after that. Or...

/^"\p{Alpha}+$/
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In the old days "\w" was marginally useful, but in a Unicode world, \p{Alpha} is probably more accurate. By the way, are you worried about words like "ain't" and "they're"?

..so how can I ensure it takes only those words which start with " not end with "..
You ask Perl to do that exactly like so:

print join "\n" => grep /"\w+?/, qw( abc efgh" "ijk lmn "opqr stuv");
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"ijk
"opqr
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