Jax vs Irelia is a wierd matchup. For a good chunk of the early game, whichever champion starts a fight is at a disadvantage. If Jax initiates, Irelia can stun him and walk away from Counter-Strike. If Irelia initiates, Jax can use counter-strike to negate her damage for most of Hiten-Style's duration.
IIRC, Jax wants to focus on farming in this matchup since he can beat Irelia in duels later in the game. In addition, it might be a good idea to get Randuin's Omen after Trinity Force for this matchup. Jax is stronger than Irelia in sustained fights, so if she couldn't blow you up quickly you can easily turn the fight and Randuin's Omen can help you stick to her while reducing her damage (Health + Armor + Attack Speed slow).
Personally, I'd have recommended rushing Phage in this matchup if possible. Irelia is a bit stronger than Jax early (she isn't quite as item-reliant) and the extra health, AD, and MS would have helped you survive and fight early.
Part of the issue some players have against champions like Irelia (and Jax, Nasus, Vayne, Vladimir, etc) is that they don't know how to properly shut-down champions early in the game or how to properly take advantage of power-spikes. They will try to take an early-game champion (i.e. Garen, Darius, Riven, Corki, Renekton, etc) against a late-game champion, and be surprised when they are outscaled after going even in lane.
I still have some issues with early aggression myself, which is part of the reason I primarily play champions who excel in the mid- and late-game rather than early-game champions. It just takes practice and experience.
You could probably get a bit more damage from Jax's kit if you altered your masteries slightly. I would suggest something like this
Jax works quite well with Blade and Spell Weaving and from the extra AP. It's a small difference, but every little bit helps.
Your runes were also pretty bad. Movement Speed Quints don't do much for you early in that matchup and AD seals are quite inefficient. I would suggest running either Armor or Scaling Health Seals (though if you take health, get one Flat Armor Quint) and using AD or Attack Speed Quints.
Oh, another note. If Jax starts with Flask, he can quite easily kill Wolves or Raptors and walk into lane for an experience advantage. You will have to consume one or two potions (of your 6) but it gives you an early experience advantage without requiring you burn your Teleport. While you have less sustain, you can easily back after a couple minutes and Teleport to lane with an item advantage. Just make sure you start with Counterstrike at level 1 if you plan to do this.
