Not immediately, but a possibility should a shutdown happen and continue.So, if Congress fails to pass a CR or budget, will the courts shut down, delaying any movement on the cases against this law through 4CA and on to SCOTUS?

The federal government is headed into a shutdown. What happens next?
While some government entities will be exempt — Social Security checks, for example, will still go out — other functions will be severely curtailed.

Ironically, two weeks according to this.The judiciary will be able to continue to operate for a limited time using funds derived from court filings and other fees, as well as other approved funding.

US judiciary can keep operating for 2 weeks if government shuts down
The federal courts will be able to remain open and continue operations for at least two weeks should Congress fail to pass legislation to avert what would be the fourth government shutdown in a decade, the judiciary's administrative arm said on Tuesday.