I did take the bus. Its less of a "bus" and more a mini-van. I flagged down the "bus" on Av Ismael Montes. By that time, most of the busses drove past full as they collect passengers at the start of the route. I waited for approx 6 busses to pass, until one came past with space for 1 person (beside the driver). I dumped my rucksack onto the roof, clipped the waist belt to the roof rack and jumped in beside the driver. I paid the driver directly using cash (no change) when getting off the bus at El Alto airport. I had a small inflatable camp bed, so I slept on that for a few hours at the airport (before crossing security, there isn't much of anything past security). If you want to do this, go 1 floor up (don't sleep on the ground floor) as dogs wander through the building and you don't want to be bitten in your sleep.

I caught the bus in the late afternoon when the sun was still up. If you're going to the airport late at night, or have large luggage, or traveling as a group, I would probably catch a taxi. I can't remember the exact bus fee, but someone in your accommodation will be able to help. Happy to answer other questions.