What is a word that means "relating to spiders" in the same way that arboreal means "relating with trees"?
My first instinct would have been something like arachnean but apparently that's not a word.
Adjective: arachnoid u'rak,noyd
(zoology) relating to or resembling a member of the class Arachnida
arachnidian, spidery, spiderlike, spiderly
I haven't been able to find a fully satisfactory word meaning "relating to spiders" that is in common use. Drew's suggestion of arachnoid and Graffito's suggestion of arachnidan are probably the two best options, if you must use an adjective. (In attributive position, it's often possible to use a noun in English, e.g. we can just say "spider habits" to mean "habits related to spiders.")
It sounds like you are looking for a word derived from Latin or Greek. I used Wiktionary to check the words in these languages for "spider." It says they are "aranea/araneus" and "arachne" respectively.
Then I used OneLook Dictionary Search to look for words starting with arane- and arachn-. Here are some candidates that I found:
araneal has apparently been used by someone at some point, but it is extremely rare. Only one of the dictionaries indexed by OneLook gives a definition: Wordnik ("Pertaining to or resembling the spider.") It's not listed in the Oxford English Dictionary, it doesn't show up on the Google Ngram Viewer, and I can't find any indication that the hypothetical Latin ancestor "aranealis" was ever used (although there is some species named Mesochorus aranealis). So even though it is the word I found that is most similar in formation to arboreal, I would be cautious about using it. However, one advantage to using a word like this that is almost never used is that it hasn't developed other conflicting specialized meanings, like the following words.
araneous also exists. However, it seems to have a specialized meaning of "resembling or full of spiderwebs" (it seems "spiderweb" was another meaning of Latin aranea). So it might not be a good choice for you. A variant form is araneose.
arachnoid, oddly enough, has the same specialized meaning as araneous. It often refers to fibrous materials that are like spiderwebs, rather than things that resemble or pertain to spiders themselves. But it can have the latter meaning. My sense is the the -oid suffix sounds more natural when used to mean "resembling" rather than "pertaining to," which biases me against using this word to mean "pertaining to spiders."
Wiktionary itself has an Appendix:Animals that suggests that arachnidan and araneidan are the adjectives corresponding to "spider. However, there are minor issues with each of these.
Arachnidan may refer to a larger class than "spiders." It is derived from the word arachnid, and may be applied to any animal of this type, which includes scorpions, ticks and mites in addition to spiders. The variants arachnidean, arachnidian would have the same problem.
Araneidan usually refers to a subset of the spiders: the araneids, spiders that belong to the orb weaver family. The variant araneidal would have the same problem.
There are also a few possibilities for adjectives derived from the English word spider. Spidery is relatively common (compared to any of the Latinate words), and generally has the sense "resembling a spider", although in some cases it seems closer in meaning to "resembling a spiderweb". Spiderish is less common than spidery, but to me it seems that spiderish has a somewhat vaguer meaning. Spiderlike is an adjective that more explicitly conveys "resembling a spider (or part of a spider)". Spiderly seems to be more or less a synonym of spidery.
Without exposing to much ignorance, I feel saying "Arachniphile" would be a justifiable construction and a generally translatable insertion into conversation. "Philia"(Arachnaphile) being a strong tendency toward and "Phobia"(Arachnaphobic) being an aversion to, they appear to be suffixes in opposition.