In the event of a government shutdown, special counsel Robert Mueller's office would *not* be affected. They're funded through a "permanent indefinite appropriation," per a spokeperson
As for the rest of the federal judiciary, a spox says they project the courts would stay open for ~3 weeks using other sources of funds — fees and no-year appropriations balances. But they stress they'd still look for ways to cut back on things like travel and training
During the 2013 shutdown, you may recall there was something of a rebellion by judges — a number of chief judges declared all personnel "essential" because they were frustated that the administration of justice was getting caught up in the budget stalemate law.com/nationallawjou…
If you're curious, here's the memo that went out to federal courts on Wednesday about the possible shutdown. The funding that's available for ~3 weeks would also keep federal defenders, probation and pretrial services, and federal bankruptcy courts open 
To recap: If there's a government shutdown, the special counsel's office will keep working, and the courts will be open
