I am sorry I was trying to move my post from Answer to Discussion but I can't quite figure out how.
I am a native Japanese but have never heard of the word, 学徒大工. Yes, 大工 means carpenter and 学徒 means student but in general 学徒 refers to younger students or it is often associated with terms used for students' roles during WWI and WWII such as 学徒隊、学徒兵、戦没学徒、学徒出陣.
I looked up this word, 学徒大工, online but nothing came up besides this very question on this site and a bunch of Chinese sites. I think it is best to stay away from using this word.
Am I correct to gather that when your apprentice period is over, you would like to give a letter of appreciation to your teacher/master?
If so, when we sign our name at the end of a letter, one thing that we do sometimes to show our respect to the receiver of the letter is to add 拝 at the end of our name. So if your name is Patrick Monk, you would sign, モンク・パトリック 拝.
However, to do this, the rest of your letter should somewhat follow the typical format of Japanese letter writing (please see
http://www.sanseido.net/Main/Words/hyakka/howto/23.aspx).