Kuukausittain merkitsen jokaisen menon vihkoon, jotta budjetti pysyy tasapainossa.

Finnish often drops subject pronouns when the verb ending already shows the person. Merkitsen is the 1st person singular present form (I mark / I write down), so minä is usually unnecessary unless you want emphasis or contrast.

Kuukausittain is an adverb meaning monthly / on a monthly basis. The ending -ttain / -ittäin is a common adverb-forming suffix used to express frequency or distribution (roughly “per X” or “in X-intervals”).

• kuukausi (month) → kuukausittain (monthly)
  Similar patterns: päivittäin (daily), viikoittain (weekly).

Yes, and the meaning is very close. The difference is mostly style:

• kuukausittain = monthly, slightly more formal/summary-like
• joka kuukausi = every month, a bit more concrete and conversational

Both work in this sentence.

Jokaisen menon is in the genitive singular:

• jokaisen = genitive of jokainen (each)
• menon = genitive of meno (expense/outgoing)

Here it functions as the object in a “complete/definite” sense: you are marking down each expense (in full, one by one). In many contexts, Finnish uses genitive/accusative-type object marking when the action is seen as complete for each item.

Jokaista menoa (partitive) is possible, but it changes the nuance. Roughly:

• merkitsen jokaisen menon = I record each expense (treating each one as a complete item)
• merkitsen jokaista menoa = I’m recording expenses in a more “ongoing/indefinite” way, or focusing on the process (less “one-by-one completed item” feel)

With jokainen (each), genitive is very common because it naturally points to complete individual items.

Vihkoon is the illative form of vihko (notebook). The illative answers “into where?” and often corresponds to English into:

• vihko (notebook) → vihkoon (into the notebook)

The illative can look different depending on the word type; for vihko, the illative is formed with a lengthened vowel plus -n (so you get -oon).

That would change the meaning:

• vihkoon (illative) = into the notebook (you enter/write it in)
• vihossa (inessive) = in the notebook (location/state: it is already in there)

With the action merkitsen (I write/mark down), vihkoon is the natural choice: you’re putting information into it.

Because jotta budjetti pysyy tasapainossa is a subordinate clause (a purpose/result clause). Finnish normally uses a comma before subordinate clauses introduced by conjunctions like jotta, että, koska, etc.

Yes. Jotta introduces a clause expressing purpose (and often also the intended result):
... jotta budjetti pysyy tasapainossa = ... so that the budget stays in balance.

In Finnish, jotta is the standard choice for this kind of “purpose” relationship.

Pysyä means to stay/remain, so it emphasizes maintaining a state over time:

• budjetti pysyy tasapainossa = the budget stays in balance (continues to be balanced)
• budjetti on tasapainossa = the budget is in balance (state at a point in time)

Because the first clause describes a repeated habit (monthly), pysyy fits well: the habit helps the balance continue.