This page provides details on how to find four-socket Monarchs in Diablo 2 Resurrected.
Finding runes is relatively straight-forward, either the monster or container you score loot from has a high enough Monster Level (mLv) or is located in a high enough Area Level (aLv) to generate a rune, or it’s not. General rule of thumb, the higher the difficulty/later the Act, the higher ceiling for rune drops.
Socketed equipment functions on a similar principle, but it can be less obvious, as the correlation between number of sockets in an item and rune rarity aren’t as clear. You can buy 3-socket weapons and shields in Normal difficulty, Act 2, so it may be reasonable to expect 4-socket weapons and shields to drop shortly thereafter. This, however, is sadly not the case.
Both 4-socket one-handed swords and 4-socket shields are desirable early-game primarily for the Spirit runeword, which has similar properties when created in a sword or a shield, namely +2 to All Skills, +25%-35% Faster Cast Rate, +22 Vitality and +89-112 Mana. With a relatively low level requirement of 25, you’re just not going to find anything better at the time, or in the cases of shields, perhaps not ever. Since the highest rune requires is Amn, it’s also not unlikely that you’ll find one just by playing through Normal difficulty, but failing that, Nightmare [Countess] will regularly drop them.
Getting the socketed base item, however, can take much longer, depending on your luck.
The maximum number of sockets an item can have is based on its Item Level (iLv) an invisible stat that’s usually the same as the mLv of the monster that dropped it (or the aLv of the container). In the case of 4-socket shields, your options are limited. If you’re a Paladin, great! You can find a 4-socket Paladin shield relatively easily and early - even the small (2x2) shields can have 4-sockets, given a high enough iLv! Make a Spirit out of the first one you find, as the runes are cheap enough to waste. Ideally, however, you’ll want to find a shield with high (50%+) base Block Chance and a base +20-30 to All Resistances.
If you’re not a Paladin, however, this is going to be a more painful process. There are only three shields that can spawn with 4-sockets, the Monarch, Aegis and Ward, and of the three, only the Monarch has an even remotely reasonable Strength requirement (and even then, it’s a hefty 156). Worse yet, these are all elite shields - you won’t likely find one until Hell difficulty.
The only solution for scoring a Monarch is volume-killing. We found one in the [Mausoleum] and another in the Catacombs in Hell difficulty, but Hell Cows are arguably better choices… if you can stand putting off such a vital piece of kit. Loot weapon racks, kill enemies (especially the minions of Unique monsters) and if you’re serious about getting a Monarch, keep your Magic Find low - finding a magical Monarch just means you missed out on a potential base for Spirit.
Once you do find a Monarch, ideally it’ll be a natural 4-socketed specimen. This is, unfortunately, incredibly unlikely, and the first worthwhile Monarch shield you’ll find will probably be an unsocketed specimen. In this case, you still have options - you can take a Monarch to [Larzuk] in Act 5 and, assuming you’ve completed his quest, you can have him add sockets to the Monarch, which should yield a 4-socket specimen. Alternatively you can use the following [Horadric Cube] recipe to add sockets to a Monarch:
Tal + Amn + Perfect Ruby + any Shield = socketed Shield
The formula above will add a random number of sockets into the shield, based on the maximum number of sockets said shield can possess and the shield’s iLv. In the case of the Monarch, any Monarch that drops can have up to four sockets, and has a sufficient iLv to carry said sockets, so this isn’t much of a concern. Note that you can only use normal-quality shields in this formula, low-quality, superior, socketed, magical or rare shields will not work. Also, since the number of sockets added is random, expect this to take a few attempts before you get the 4-socketed Monarch you need.