We are arriving at 08:35 in Madrid on Sunday, September 24, on a Delta/Air France flight and need to catch a bus to Bilbao at Avenida de America. I need to know how much time we will need to get through all the customs, emigration, etc. and then get to the bus station, so I can book tickets on the bus and not be stressed about getting to the bus station in plenty of time. We are not Spanish speakers and realize that everything will take longer than we expect. Thanks for your help!
You should wait til you get there to book a bus to Bilbao, because you may not arrive at 8:35 if the flight is delayed.
Are you meeting someone or a tour group at Avenida de America?
If not there is bus service the same company Alsa to Bilbao from Madrid Airport Terminal 4
https://www.alsa.es/en/our-destinations/airports/madrid-barajas-airport
ALSA seems to have buses nearly hourly. It's impossible to predict how much time you'll need with good reliabilty, so I'd plan to buy the ticket upon arrival at the bus station in Madrid. It's unlikely a mid-day bus will sell out, and this way you won't miss the bus you're ticketed on or end up cooling your heels at the bus station for a couple of hours if everything goes swimmimgly at the airport.
I don't know how much a taxi from the airport to the bus station would cost, but I'd tend to go that way (if the airport departures don't fit your schedule). Your public-transportation options involve transfers, I believe. With luggage and (I'm guessing) after an overnight flight, simple would be better than complex.
Your lack of Spanish should not be a problem. There are ticket machines, and they can be set to display prompts in English. You'll be specifying that you want a one-way ticket, choosing the date and then choosing the time. I don't remember whether you have an option to specify two tickets in a single purchase, but I imagine you do. The machines were happy to take my US credit cards.
If you prefer not to use a ticket-vending machine, there will be a staffed counter available. That will probably take longer, and I think by policy all ticket sales stop a specfic number of minutes before the bus is scheduled to depart. There's a good chance the agent will speak some English, but I always have my destination written out in all capitals, along with the travel date (24-09 in your case). You'll be asked whether you want a round-trip ticket (ida y vuelta) or just one-way (ida).