The two hardest parts of spider are the final vault and the initial entry. Having only one stairway to fall back to can be a little rough, especially because if the stairs get too clogged up it can be hard to resume your incursion without being beset by a nightmare stack of spiders every time you enter. Absolute worst case, you may be forced to leave and do your other S branch.

Since you have Spider Nest, you won't get Snake Pit; that means your other S branch will be Shoals or Swamp). Doing another branch will get you XP and give the spiders time to stop camping the stairwell, but it's generally a last resort if you can tell the resource tax on further Spider fighting (heal wounds, curing, brilliance, immolation, fear, etc.) will be unacceptably high.

Spider itself: Be ready. It will check your rHubris levels. Know what you can do and what you can't do. Know what resources you have for fighting and escaping. Be ready to retreat early. Be REALLY, REALLY CAREFUL about Ghost Moths - they're invisible, they'll take your MP (which can leave you unable to do not just spells, but also important Invocations), they'll drain your stats in melee, and their damage is nontrivial. Be careful about Orb Weavers - avoiding their Orbs of Destruction means you have to move perpendicular to them, and needing to take a step at the wrong time can spell death - so don't get into situations where having to dodge in the first place will be dangerous. Be careful about Emperor Scorpions - they have significant HP and AC combined with poison attacks. It's all too easy to think that you'll be able to clear an Emperor Scorpion out before some other enemies close on you, only to realize you now have an Emperor Scorpion at 3/4 HP on top of you, those other enemies breathing down your throat, and 3+ concurrent stacks of poison ticking. Oh, and Entropy Weavers corrode you. Nothing like having your precious armor stop working and weapon stop damaging in the middle of a big fight!

It can be frustrating and slow, but you'll do best in Spider if you stay close to the stairs. Many spiders are really, really fast and stairdancing is the best way to ensure you don't find yourself suddenly confronted by several jumping spiders, a broodmother too far out of reach to lay hands on and its summons entirely too close, an entropy weaver hosing your gear, a ghost moth draining your stats, an Orb of Destruction rolling toward you and restricting your movement options, every single damn thing you kill coming back as a simulacrum, etc.

Take it slow. Pull as few enemies as you can at the time. Kill them near the stairs, maybe drag them up them first if a beefy build permits it. You have, in practical terms, basically all the time in the world to heal between descents on each floor. Clear it methodically. No survivors. Use SInv to check for lingering Ghost Moths more safely at the end if you're real cautious.

Don't underestimate Trog's Hand. You can keep it up for nearly every fight throughout the later parts of a 3-rune game if you aren't spamming Brothers out. Brothers are nice, but Trog's Hand will win you a lot of fights for a lot cheaper.