Leam Chabang is a port that is really short of information, with the same questions being asked, and hardly ever answered. I grabbed a picture of the Taxi prices hoping it would help someone down the track.
We arrived in Laem Chabang on Dec 22 with NCL Jewel. I had read about a mall shuttle service also, but there were no obvious signs for a shuttle service. Others we met on the cruise had issues finding a Grab/Bolt to Bangkok, I don't know of anyone who tried for Pattaya.
There is a taxi Mafia that operates from the port with fixed prices.
USD40 (1,400THB) to Pattaya.
We originally booked our transfer with Thai happy taxi, but a couple of weeks out they increased the agreed price by 1000THB and offered a small car instead of the SUV we had been promised, and insisted on us sending them a deposit.
We ended up booking Jojo's taxis for a bit of a tour and transfer to Bangkok (I'm not allowed to post a link here), and it went well, happy with the car and driver.
NCL did offer Transfers to the Airport for USD$50 per person, but nothing for Pattaya.
Hope that helps.
Pete.