Yes, when you use an apostrophe to show missing letters that have been contracted out of the beginning of a word for whatever reason, you still use a capital first letter for the sentence, meaning you'd use a capital A. A common example we see is in answer to a question of why when someone answers with a form of because where they've omitted the be at the beginning.
Examples from published works:
'Cause I don't think he go to Placerville or Sacramento.
-Corbett Mack: The Life of a Northern Paiute by Michael Hittman, pg. 144
'Cause I don't wanna be involved in nothin' that goes on around
-Getting Played: African American Girls, Urban Inequality, and Gendered Violence by Jody Miller, pg. 62
'Cause I don't care.
-Misadventure: Monologues and Short Pieces by Donald Marguiles, pg. 76