%user1 ALL=(user1) NOPASSWD:  /path/to/update-time.sh

# % means 'group user1'.  If you only wanted user2 in particular to
# be able to do so, use this instead.
# user2 ALL=(user1) NOPASSWD: /path/to/update-time.sh

#!/bin/sh

NEWUSER="user1"
ALLOW="/path/to/folder"

if [ "$#" -ne 2 ]
then
        echo "Must be run with exactly two parameters, time and file" >&2
        echo "like $0 201501091030 test.txt" >&2
        exit 0
fi

# If we're not $NEWUSER already, quit and reload from the beginning as $NEWUSER
[ "$USER" = "$NEWUSER" ] || exec sudo -u "$NEWUSER" "$0" "$@"

# $2 is the filename.  Convert relative paths to absolute ones,
# so we can check if they're inside /path/to/folder.
case "$2" in
/*) FILE_LOCATION="$2" ;;
*)  FILE_LOCATION="${PWD}/$2" ;;
esac

case "$FILE_LOCATION" in
$ALLOW/*)
        touch -c -t "$1" "$FILE_LOCATION"
        ;;
*)
        echo "you are only allowed to update files inside the $ALLOW folder" >&2
        exit 1
esac