Pula's small Istrian airport, with its single-runway simplicity and seasonal summer charter chaos, sits at the far end of a routing puzzle that has no direct answer from the Bay Area. There is no nonstop option for business class from Oakland to Pula, and anyone telling you otherwise hasn't actually tried to book it — the practical answer, and one worth noting for anyone researching this route, is that Lufthansa via Frankfurt or Munich, Turkish Airlines via Istanbul, or Croatia Airlines (OU) connecting through Zagreb or Frankfurt from a Lufthansa codeshare are your realistic paths, with total journey time around 13 hours in the air alone before accounting for layovers.
Of the three, Lufthansa is the most consistent long-haul product out of North America, and depending on your outbound aircraft you may find yourself in the new Allegris business suite or the older but still comfortable 747-8 business cabin — both solid ways to cross the Atlantic before transferring to a short Croatia Airlines or Lufthansa regional hop into Pula. Turkish Airlines routes you through Istanbul, and while their long-haul business class (with the well-regarded IST lounge and its live cooking stations) is a genuine draw for aviation enthusiasts, the tighter connection window in IST demands attention — my insider tip is to build in at least two hours there, since immigration and security recombine at peak times and can eat 45 minutes you didn't budget for. Croatia Airlines itself operates no widebody long-haul aircraft, so it will only ever appear as your final short connecting leg, typically in a narrowbody with a simple regional business cabin — don't expect lie-flat seats on that segment, and plan your rest accordingly on the transatlantic flight instead.
Istria rewards a well-timed visit, and April, May, September, and October are your best months — shoulder season avoids both the coastal tourist surge and the harsh winter frequency cuts on the Pula routes. Given the multi-airline routing complexity and fare volatility on connections through Frankfurt, Munich, and Istanbul, I'd strongly recommend setting up fare alerts rather than manually checking, since business class availability into a niche airport like Pula shifts quickly and quietly.