What is the most efficient way to update a product table with 1 million rows in MySQL
What alterable properties of the product table e.g. Indexes might slow down 
  updates? 


What do you care? It is totally irrelevant. See, by your own words:


  For each row, I need to perform an expensive http operation and then update 1
  field in the row based on a returned header.


So, all the MySql performance wont matter - unless you run that on a mobile phone - because you are going to be controleld in speed by "an expenive HTTP operation".

Make a queue (can bedb based), request the http stuff one by one, run updates. THe updates wont take long. Get into multi threaded programming and use some hundred threads - each running in parallel - to get some http speed.

But the update performance in MySQL will not matter unless you want to wait untill you got all data, which may work or not - normally I would say it is not an optimal way.