Hi and welcome to LinuxQuestions!
Most likely the modification time of your logs is the same as the date in the file name, therefore you can use the find command with -mtime (not -mmin as in your example):
find /home/sarathi/sarathi -mtime +10 -delete
warning: test this command without -delete to prevent unwanted deletions, until you're sure of the result of the find command.
If you still want to remove older files based on the date in the filename, you can take advantage of the fact that the names are sorted alpha-numerically. This is because of the yyyy-mm-dd format of the date (it wouldn't be true for example if you had dd-mm-yyyy), therefore something like:
for file in sarathi_????_??_??
if [[ $file < $(date -d "10 days ago" +sarathi_%Y_%m_%d) ]]
zip -g archive.zip $file && rm $file
should do the trick. Note the -g option of the zip command adds a file to an existing archive and prints out a warning if the archive doesn't exist yet, but it should be created properly.