The cheapest trip is to use public transportation. You catch the "LOT C" bus outside of your arrival terminal at LAX. From LOT C, walk to the LAX Metro bus terminal, and catch the "232 Line" bus, which will go from Lot C to downtown Long Beach. However, you will need to "transfer" (costs about an extra 10 cents from the 232 bus to the 445 when the bus stops at the intersection of "Pacific Coast Highway (Hwy 1)" and the Harbor freeway (110 freeway). You need to get up to the busway at the upper freeway level (they have elevators, I think). Ride the 445 to "San Pedro Park N Ride Lot II, which is across the street from the cruise terminal.
Here's the description from the Metro's website. Don't pay attention to the actual times, as my search just happened to be for the next available schedule when I did the search. The 232 bus runs about every 15 or 20 minutes during weekeday, from about 6:00 am until 6:00 pm. It is less frequest on weekends and evenings.:
You are traveling on Thursday and Arriving at 2:21PM . Fare class: Regular. Max walk: 1/3 Mile. Mode: ANY.
Starting at LAX AIRPORT
Ride Metro Local Line 232( LONG BEACH TRANSIT MALL) heading west
From: LAX TRANSIT CENTER DOCK 8 Lv: 12:57PM
To: PACIFIC COAST/FIGUEROA PL(SW corner) Ar: 02:05PM
Pay $1.25, Monthly Pass: $62.00, (EZ Pass accepted)
Ride Metro Express Line 445 (SAN PEDRO - PACIFIC-21ST) heading south
From: HARBOR FRWY/PC HIGHWAY STATION Lv: 02:23PM
To: HARBOR BEACON PARK N RIDE LOT Ar: 02:29PM
Pay $1.85, Monthly Pass: $80.00, (EZ Pass accepted)
Ending at SAN PEDRO II PARK AND RIDE
Total cash fare = $3.10 *For this trip consider a Metro Day Pass
Trip time is about 1 hour and 32 minutes.
Trip distance is about 20.61 miles.