If you mean size as in "bytes", try head() from LWP::Simple, and check out the content-length — second value when called in list context.

use LWP::Simple;
local $\ = "\n";  # add newline to print
print +(head('http://perlmonks.org/images/monk1sm.gif'))[1];
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If you mean the image dimensions, you'll have to download the file, for example using get() or getstore() from LWP::Simple, and read the dimensions with Image::Size or Image::Info.

Here I use Image::Info because it doesn't require the image to be stored as a file — just use a scalar reference to the raw image data:

use LWP::Simple;
use Image::Info qw(image_info dim);
my $raw = get('http://perlmonks.org/images/monk1sm.gif');
printf "WxH: %dx%d\n", dim(image_info(\$raw));
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