My approach with Japan the two games I’ve played has been to build IC in India and FIC, build Arm there, and build TRNs also to keep a steady flow of Inf from Japan to Persia. That puts a lot of pressure on Cau to allow either Germany or Japan to take it. Once you have a big Axis presence in Cau, that Sink IC becomes less valuable to the Allies.
In other words, go around it - to the south. Germany can take Cau a lot sooner if Japan is in a position to immediately reinforce with a bunch of stuff.
My tourney game I also faced Russian defence (but no IC) and was also not able to take Sink / Yakut until J5 or so, but still there was enough money that I had 2 ICs and 7 TRNs running. I think Japan’s priorities should be building TRNs, taking Australia / New Zealand / India ASAP (J3) and picking up all the island Inf. Japan’s TRNs practically pay for themselves because of the island Inf supply, and because they increase the value of your forces by moving them to the front much faster.
Haven’t actually faced that American IC in Sink though.