The cone referenced is
Yoyogi-koen, as hinted at in the hint. This is in the bottom-left quadrant of the map along the green (Chiyoda) line, just above Shibuya, which is in large, bold text. Harajuku is nearby (to the right of Yoyogi).
Take two stations east along the
green (Chiyoda) line to Omote-sando, which is a station containing both the green (Chiyoda) and purple (Hanzomon) lines.
Two more stations along the purple (Hanzomon) line, one north-ish, one southeast-ish, and you're at
the Nagatacho station, the final answer.