About half way through Episode 04: Double Vision you'll acquire the spray bottle from Emmett's lab.
Upon returning to the Expo entrance outside (where Doc, Edna, Trixie, Artie, Guy and Cueball are), equip the spray bottle and interact with the DeLorean to stash the item. Interact with the DeLorean again to find out if the chronometric analyser light has turned green. If it hasn't, just wait a minute and you should be fine. Interact with Doc and use the option "The light is green". Doc will take the DeLorean on a test run to examine the issues with the time circuits. Upon returning, he will tell Marty that he has been away for roughly 9.5 hours.
Wait another minute or so and then interact with the DeLorean again. Once the light has turned green, repeat the same steps as above by telling Doc. Upon his return, the achievement should pop as it has been more than 12 hours since you stashed the spray bottle.
NOTE: If you forgot to stash the spray bottle inside the DeLorean the first time round (like I did), don't worry. Be sure to interact with the DeLorean with the spray bottle and follow the steps above (i.e. check the light, tell Doc, etc.). I think the game needs you to have aged that spray bottle by 12 hours for story progression so theoretically you might have unlimited DeLorean time tests. At minimum, you'll have three, which should be plentiful for getting this achievement!