Sulla mensola c’è anche la cassetta degli attrezzi, nel caso serva presto.

Sulla is the contraction of su (on) + la (the, feminine singular). In Italian, prepositions often merge with definite articles:

• su + la → sulla (on the, f. sg.)
  You use sulla because mensola (shelf) is feminine singular:
  “Sulla mensola” = “On the shelf.”

C’è comes from ci + è:

• ci here is an expletive/adverb meaning “there”
• è is the third-person singular of essere (to be)
  The apostrophe shows that the vowel i of ci and the vowel e of è merge:
  ci + è → c’è (“there is”).

Anche means “also” or “too,” indicating an additional item. In this sentence:

• Placing it before the noun phrase (“c’è anche la cassetta…”) focuses on “the toolbox, too.”
  You could also say “c’è la cassetta degli attrezzi anche,” but it’s less common and slightly more emphatic at the end.
  Position changes the nuance, but both convey that the toolbox is an extra thing on the shelf.

Here degli is di + gli (of the, masc. plural):

• attrezzi is plural (“tools”), whose article is gli
• di + gli → degli
  So cassetta degli attrezzi literally = “box of the tools.”
  You could hear cassetta di attrezzi in a very general sense (“a toolbox”), but using degli implies “the specific set of tools” and is more natural for “toolbox” as a fixed expression.

After expressions of condition or possibility like nel caso (“in case”), Italian uses the subjunctive to convey a hypothetical event.

• nel caso (che) serva = “in case (it) might be needed”
  The subjunctive serva shows that we’re talking about something that may or may not happen.

Yes. In spoken and even in many written contexts, you drop the conjunction che after nel caso:

• With che: “nel caso che serva presto…”
• Without che: “nel caso serva presto”
  Both are correct; omitting che makes the phrase more concise and idiomatic.

Presto is an adverb meaning “soon” or “quickly.” It modifies serva (“is needed”), so it naturally follows the verb:

• “serva presto” = “is needed soon”
  Placing presto at the end keeps the verb-adverb order typical in Italian. Knowledge of this helps you position adverbs after the verbs they modify.