It has to be a personal reference/inside joke between Tesreau and Criger. Sticking was a commonly term in the early 1900's (still used today though less so) describing someone's ability or when used in a derogatory or ribbing fashion their inability to hit.
The phrase was even used by Louis Masur in his book Baseball's First World Series on page 18 when he speaks about the hitting ability of Eddie Phelps in relation to Criger...... "adherents dismissed Young and Dinneen as being more lucky than skillful, and they trumpeted their catcher Eddie Phelps as a superior "stick"- man to his weak-hitting counterpart "Lou Criger".
Tesreau was kind heartedly razzing Criger using the common terminology of the day.
The only other thing it could be was when Jesse Burkett let Criger use his favorite bat/stick that Criger subsequently broke. From the Sacramento Union in 1908...He told Lou that he would let him use the famous stick for a few days until it had taught him how to hit, for he claimed hi* bat had that virtue, but made the catcher swear that he wouldn't hurt it and that he would return it before the big league race became close. One day he told his teammates that he wished he had his bat back, as he had a hunch that some harm was happening to it, and he declared that he was going to write for it that night. The same day that Burkett had the hunch Criger broke the famous stick so that it couldn’t be repaired. Here's the link...
http://cdnc.ucr.edu/cgi-bin/cdnc?a=d&d=SU19080809.2.159
But I still think the reference is a general one relating to his inability to hit.