What is the fastest way to obtain Melmetal without using Rare Candy or buddying?

Without Let's Go, you can only obtain one Meltan through the special research. With Let's Go, you can get a Mystery Box to catch Meltan. However you only get a limited number of Meltan and I don't have a Let's Go game, though my friend does. Whilst you can use Rare Candy or buddying, that could be spent to get candy for other Pokemon.

What's the fastest way to get the required 400 candy to evolve a Meltan into Melmetal without using Rare Candy or buddying?

With the special research, you can get 8 candy - 3 candy from catching Meltan and 5 candy from completing the research.

Transferring Pokemon to Let's Go lets you open a Mystery Box. If you don't have Let's Go, you can transfer to a friend's Let Go account. After opening it, 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 walking doesn't change Meltan's rate of appearance; all 20 Meltan can appear without any movement.

Use Pinap berries. You will then obtain up to 120 candy from catching and 140 after transferring them. Apparently Meltan rarely flees unless you use Pokemon Go Plus or the app crashes, if it does even flee otherwise, making it safer to use Pinap berries over Razz berries in case Meltan breaks out of the ball.

It's possible to ensure that all 20 Meltan appear. The first should spawn at about 29:53 time remaining. Afterwards they spawn 1.5 min after clicking 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, open your character profile at 28:23. A timer for multiple 1.5 min periods can be helpful such as the built-in Google app or a pomodoro timer app.

After opening the Box, you'll need to wait 3 days, decreased permanently from a week as of 24th April, and then retransfer a Pokemon to a Let's Go game before opening it again. The 3-day timer starts from when you open the Box. After opening the Box again after 3 and 6 days, that can total to 420 candy. This is in addition to the candy from the special Meltan quest. However, without Rare Candy and buddying, you are reliant on a Let's Go game, either yours or a friend's.

Assuming that you haven't already started, and excluding the special research, it can take as little as 6 days and 30 minutes by opening the Box 3 times; at the start of the 6-day period, midway after 3 days, and after a further 3 days with 30 minutes for catching all the Meltan after the last box. This time can decrease during special events. For example a bonus that doubles the candy gain on capture would lower the time to 3 days and 30 minutes since you would only need to open the Box twice.

[Current bonus] The Meltan Box lasts 1 hour instead of 30 min. This effectively doubles the candy you can get, and means to get Meltan, you only need to open the Box twice instead of three times. This is not an event, but is part of the bonuses provided in response to COVID-19.