With public transportation the usual tradeoff is time versus money. If you have all day long to travel 40 mi/64 km, it can be done entirely with MTA and OCTA buses. I suspect you do not want to take all day, so adding a rail or taxi segment will help move things along.
John Wayne/Orange County Airport is quite a distance from DTLA and, like most of Orange County, not well-served by public transportation. Overall, it will not be a great trip. Using the 511 service from Pershing Square to John Wayne, my results were all around 2½ hours from start to finish, each way. A private airport shuttle service like SuperShuttle or PrimeTime would probably be around $75, a taxi, somewhere north of $110, and depending on traffic would easily cut an hour to an hour and a half off that time.
Another John Wayne trick is that the two OCTA buses which drop off and pick up directly at the terminal, route 76 and route 212, are for commuters, and not particularly useful for connections.
If your friend is willing to pick you up from the airport, would s/he be willing to pick you up from a commuter rail station? There are several to choose from not far from SNA, including Tustin, Orange, and Santa Ana.
Even the MetroLink ride is a minimum of an hour from Union Station to Santa Ana, and up to an hour and a half to Tustin. You can take the Amtrak Pacific Surfliner on the same route and save 15-20 minutes or so, paying a little less than double. Amtrak only stops at Santa Ana and Irvine, and unfortunately, this section is by far the least scenic of the entire Pacific Surfliner route.
From Tustin MetroLink
You've correctly identified the Orange County Metrolink to Tustin, then Irvine Shuttle route A as the most reasonable route. It is also possible to take iShuttle Route B; while it does not stop at the terminal itself, Dupont Dr. is perhaps a 15-minute walk from the terminal. Between the two iShuttle routes, the service gap is narrowed about 9am and 2pm, and after 7:30pm.
According to TaxiFareFinder.com, a taxi ride from the Tustin MetroLink to SNA would run around $20. I do not know how reliable the fare information is, but if you call for a taxi on the train from Santa Ana, you could have one waiting for you and be finished easily. I do not know if there are taxis available on demand at Tustin.
From Orange MetroLink or Santa Ana MetroLink/Amtrak
Alternatively, you can get off the MetroLink at the City of Orange station, or off the MetroLink or Amtrak at the Santa Ana station, and take OCTA bus 59. The 59 does not go directly to the terminal, but will take you to the intersection of Von Karman Ave and Michelson Drive, from which it is a 15-20 minute walk to the terminal. The 59 runs all day long, albeit with a reduced mid-day schedule.
From Anaheim via Disneyland Express
I do not recommend this route, but you could also take the train to Anaheim and take the Disneyland Resort Express. This is a private service owned by Gray Line and unaffiliated with Disney. It costs $35 between Anaheim and the airport, the problem being that the Anaheim Amtrak/MetroLink station is not very close to Disneyland. The nearest pickup spot is at the Holiday Inn Anaheim Resorts, which is about 1¾ miles/2.8 km away, and there is absolutely nothing of interest to see on that dusty, smoggy walk except for the baseball stadium.