With the special research, you get 3 candy from catching Meltan and an extra 5 candy from completing the research.
You can transfer Pokemon to your friend's Let Go account and receive a Mystery Box. Even though it isn't your game, you can still obtain a box along with anyone else who transfer to that game. You can then open it and, over the next 30 minutes, up to 20 Meltan (increased from 10 as of 12th Dec 18) will appear at your spot. It works similarly to Incense, but you do not need to walk to obtain all 20 Meltan; they will appear where you stand.
Use Pinap berries on these Meltan. You will then obtain up to 120 candy from catching the Meltan, 140 after transferring them. Fortunately Meltan appear to rarely flee unless you are using Pokemon Go Plus or the app crashes, if it does even flee otherwise. This makes it safer to use Pinap berries instead of say Razz berries in case it breaks out of the ball.
It is possible to ensure that all 20 Meltan appear. The first should spwan at about 29:53 time remaining. Aftewards they spawn 1 min 30 sec after you clicked on the last Meltan. If one doesn't appear after 1.5 min, open your character profile and then close it, which should make Meltan appear e.g. if you clicked on a Meltan at 29:53, then open your character profile at 28:23 and then close it, a Meltan should appear. A timer for 1:30 can be helpful such as the built-in Google app or an App for a pomodoro timer to time down multiple 1:30s.
After opening the box, you will have to wait 3 days, increased permanently from a week as of 24th April, and then retransfer a Pokemon to a Let's Go game before you can open it again. The 3-day timer starts from when you open the box.
This means that through the box you can only obtain up to 140 candy a week, excluding RC and buddying. After opening the box twice more after 3 days and 6 days, that can be up to 420 candy assuming you have a good supply of Pinap Berries. This is in addition to the candy you obtain from the special quest. However, without Rare Candy and buddying, you will be reliant on transferring to someone else's Let's Go game or buying your own to get enough candy.
Assuming that you haven't already started, it should take a little over 6 days excluding events with special bonuses like double candy. You'd need to open the Mystery Box 3 times; at the start of the 6-day period, midway after 3 days, and after a further 3 days i.e. 6 days. This is 6 days and 30 minutes if you open the box asap after opening it, with the extra 30 minutes being the time catching all the Meltan after the last box. This time can decrease during special events, for example when candy bonuses from capture and transferring are doubled which would lower the time to 3 days (opening the box at the beginning and end of 3 days).