How do I kill a witch in minecraft as a beginner, no weapons, milk, or advanced items in inventory

Witches drink potions to heal itself, eg it drinks potion of fire resistance when on fire, potion of healing when hurt, etc. It also throws negative potions at you like poison, harming and slowness, so keeping a distance while combatting the witch is recommended. It should take 4 to 5 hits with an unenchanted bow to kill it, or if you really need to get close, about 7 hits with an unenchanted iron sword and 6 hits with an unenchanted diamond sword.

Also look for holes in your "fully walled" castle, as creepers might have breached it without noticing you. Furthermore, hostile mobs spawn when light level is below 7. Use the F3 menu if necessary

Witches are a rare hostile mob, however they can be killed relatively easily. First off you want to set them on fire, this immediately causes them to drink fire resistance, and while they're doing that, they can't poison or throw anything nasty at you. During this time deal as much damage as possible, keep hitting her and don't stop until she dies. Of course if you have ranged weapons like bows then use them instead as then you'll be well out of the witches range, however you will need to be able to hit her otherwise she can take a while to kill as she heals up.

Also, in terms of witches abilities they are just as mobile as zombies (in terms of navigating your defences, so they cannot fly or climb walls).

How did it get into your castle?

According to this wiki:

you can find witches in most Overworld biomes in light levels of 7 or less. Witches will spawn in the dark, usually at night. Once a witch has spawned in the dark, it can survive in the daylight.

So to avoid witches spawning in the future, make sure your castle and the surrounding area is well lit (using torches or other light-producing objects).

What can Witches do?

Witches are hostile mobs, meaning they will attack you as soon as they see you. According to the same wiki as above:

A witch is armed with splash potions such as: Splash Potion of Weakness, Splash Potion of Poison, Splash Potion of Harming and Splash Potion of Slowness.

Note that you can drink Milk to negate these potion effects if you are hit by one while fighting the witch.

Witches are also unique in they way they attack (emphasis mine):

a witch has a special attack. It will throw harmful splash potions at you causing you damage until you die.

Importantly, since they use potions, your armour will be negated entirely - this is perhaps why it so easily killed you:

Because of this, the player's armor is rendered useless, regardless of the armor's material and/or enchantments.

Finally, witches can also regenerate their health:

Witches also have the ability to regenerate health

How do I kill one?

There are a couple of different strategies for fighting witches, which can be summarized as follows:

• Use a bow - using a bow you can stay out of the witch's range, meaning you cannot be hit by the potions it may throw at you.
• Get in close and deal consistent damage - dealing high damage quickly will force the witch to use health potions instead of attacking you (as mentioned in the Reddit thread linked above). Just make sure you can deal more damage than she can heal.
• Potion of Strength - you can also use a potion of strength to supplement your attacking, and help bring the witch down faster.
• Use console commands - if you're having a really tough time, this answer explains how to use console commands to kill an enemy. You can find your coordinates using F3.