I finally started Persona 4 today, and I can say that it's just as addictive as 3... I think my game clock was at around 6 hours when I finally turned the game off >.< So far, I'm really enjoying it... Not sure if I like it better than 3 or not, yet. It's been a while since I've played the beginning of 3, but it seems like this one is harder... The boss at the end of the first dungeon took me forever to beat.
Yeah, it took me a few tries to hone my strategy. Basically, I lowered her def, raised Yusuke's attack, and railed on her with physical skills. Once the prince was summoned, I just made sure to keep him down/dizzy using ice, specifically with Chie's all enemies ice skill so as to damage the shadow too. The shadow wouldn't cast White Wall if he was down, and this way he couldn't heal her. I just had to survive her all party fire skill.
But just wait till you fight the optional boss that appears after you defeat Yukiko's shadow. 1000hp and a Rampage attack that deals ~250 damage to everyone! Took me three tries to beat him, and then only because I had a Physical Mirror that reflected the 250 off the protagonist onto him, and had boosted hit/evade on the rest using "Sonic something" so two of them dodged. Additionally, I kept his def lowered and raised my entire party's attack twice with two Battle Calls. Also, using Yosuke's Sonic Punch helped, because it has a relatively high chance for critical hit, especially with boosted hit/evade. This downs the enemy and allows you to follow up with a rush.
Anyways, the one thing I don't like about the battle system is that if the protagonist dies you automatically lose. You can't give him a Revival Bead or anything. It's retarded, imho.
Oh, eventually you will get to a holiday where the protagonist says he's going to "go somewhere you don't usually go". I strongly recommend going to the shrine, because you'll meet the fox there. He can restore your SP in the TV world, allowing a virtually indefinite stay if you desire it. And don't forget to study for midterms!

(I got a top 10 result)
*goes back to fighting the super-gay shadow in the bathhouse dungeon. *