1. We will win the game against Japan.
2. We will win the game with Japan.
Q1. Are both the same in meaning?
Q2. What is the part of speech of 'against Japan'? Is it an adjective phrase or an adverbial phrase?
They could mean the same thing, but to me, the first sentence is clearly talking about a game between us and Japan, while the second could be that or a game in which you and Japan are playing someone else.
"against" is a preposition, so "against Japan" is a prepositional phrase describing "game" -- since "game is a noun, the prep phrase is adjectival.