If a guy walked into a sting and got arrested, what's the timeline for the next few hours?

Let's assume this occurs at 1:00pm on a weekday, is his first offense, and he had not previously retained a lawyer?

What would happen first, next, last, and roughly how long would the "processing" take?

If it was a weekday you would probably be arraigned the next morning.  It really depends on where you are as state and local laws vary. It is conceivable that in some areas you could be released as soon as someone comes to bail you out.  In either case the first part of the journey involves a ride in a police car and a short stint behind bars. Realize also that your vehicle will probably be impounded. Again, local ordinances vary.  You may not be able to get it back as quickly as you would like as it is evidence in a possible criminal investigation.

The smartest approach here is not to wind up in a room with LE to begin with.

The sooner you call a lawyer, the better.  This advise was given me by a lawyer.  I believe this is true for all states.

LE legally has to hand you your cell phone if you have a lawyer's number saved in it to call.  

While you're on the phone with this lawyer, you can tell him who to contact for bail, which in turn will let someone know you're ok, but in trouble...  

In some cities, they can hold you for quite awhile during processing before you can call anyone.  Usually HOURS later.  Also, in some cities, they can hold you for 48 hours or longer before you're before a judge to determine bail.  It all depends how good your lawyer is.

depends on how many they've busted that night and what system you're going through.  You'll be longer in an LA bust then you would in an Allentown one for example.

There are a lot of "it depends" in the answer to your question.  Is the sting in a small town or a big city?  Does LE want to make a "statement" by the arrests or not?  Is the jail full or not?  Are they practicing catch-and-release or do they really want to hold you in the cooler?  Do they have field processing van or are they going to haul you off to jail for booking?

What is certain is that you will be placed under arrest, cuffed, searched, your car's contents "inventoried" (searching by another name), your car probably impounded (if you were driving), photographed, fingerprinted....and then?  You could be kicked lose on your own recognizance, allowed to post bail right then and there, or held over to see a magistrate.  

Processing time depends upon the line ahead of you.  If could be from 15 to 30 minutes to hours.  Processing in a big city jail could mean a long night and some new friends you might not want to have.

If it is an organized sting, the press may well be involved.  If LE is going to spend the time and resources they generally want some glory.  Where I live the local LE ran a curb side sting with a copchick dressed up all nice and pretty.  They had a processing van in a nearby motel parking lot and reeled the johns in one after another.  The press was on hand for the fun.  I read about one of my neighbors getting collared.  

If you have a "name" in your community, then the local newspaper may want to have some more fun at your expense.   "Local hot-shot arrested...."

Like mother said, "An ounce of prevention is worth a pound of cure."