Only those in which he heard all the oral arguments. Let's say he misses 95% of your case, you probably don't want that justice to vote on your case if he didn't hear all the arguments.
When the Court convenes on the First Monday of October, they will proceed to consider the docket for the year. The oral hearings are months away. Until those happen, the briefs must be prepared, the Chief Justice must assign each case to a Justice, and then the Justices must consider the written briefs. Kavanaugh will participate.
The Docket for October can be found here:
https://www.supremecourt.gov/oral_arguments/argument_calendars/MonthlyArgumentCalOctober2018.pdf