The technique Nobusuke Tagomi uses to repair the cup is called kintsugi ("golden joinery") using lacquer mixed with gold dust. As a philosophy, it treats breakage and repair as part of the history of an object rather than something to disguise.
Nobusuke Tagomi's (Cary-Hiroyuki Tagawa) lack of memory of past events in the alt-world can be played off due to his life in the alt-world. His alt-self is an alcoholic and he can use this fact as a basis for memory loss.