Roughly when is a good time to battle Skeletron?

I've got full Shadow gear, a Phaseblade, 360 health, at least 10 mana stars, Starfury, Red Balloon, and other magical items.

I mostly acquired them from floating islands, but I can defeat the Eye of Cthulhu easily, and I have defeated Eater of Worlds (not nearly as easy as Cthulhu), so they have provided some decent drops.

I've just started poking around in the Underground Jungle.

Am I OK for Skeletron or should I gather Meteorite gear and/or Hellforged gear?

I'd say go for it.

You might need a decent amount of health potions, but you have enough maximum health and a good armor to withstand his attacks. There's really not a lot you can do to dodge his attacks without Rocket Boots, especially the arms.

Obviously, ranged weapon are best. I recommend the Demon Bow and the best arrows you have. As a melee backup weapon, something with a very long reach is preferable (e.g., the Phaseblade or the Blade of Grass), but you should really pack enough arrows to make that unnecessary.

If you decide to brave the Underworld before doing battle with him, replace the Demon Bow with the Molten Fury and add the Flamarang as a backup weapon. I did so too, but I found surviving in the Underworld long enough to get a decent amount of Hellstone a lot more difficult than the Skeletron encounter. After the Dungeon, it might be more manageable if you found a Blue Moon.

Although, risking your hide just long enough to gather the 45 Hellstone required for the Molten Fury might just be worth it, since it does a lot of damage — more than double of what the Demon Bow does. Plus, you won't have to waste any torches on Flaming Arrow crafting.

Depending on how well you avoid Skeletrons hands, I would say you aren't ready to battle him.

From my own experience I find it easiest to defeat Skeletron using a Flamarang, because his head (which is the only part you need to kill for him to die), is flying around a lot. And so using melee weapons is not favorable.

Fighting Skeletron at daybreak will result in him dealing 9999 points of damage to the player. So make sure to end the battle before that. (Skeltron can only be summoned at night time.)

Basically what you need is a decent ranged weapon and a lot of potions. But if you want to go melee while wearing Shadow Armor you can use Muramasa and two Feral Claws for maximum attack speed.

I had full shadow armour and a space gun. The first time I tried to beat him I died, but the second time I succeeded. Rocket boots are really good too. I only had 300 health. He's easy.

I killed him using a demonite bow and thorn chackram with full silver armor and about 2 or 3 potions ( it was just me fighting ). It depends if you know what your going up against, because I only defeated him on my second try. But to be honest you don't need much to defeat him ( atleast on ipad version ) so I would say go for it! Good luck