Este mes el alquiler sube un poco, así que debo ahorrar más.

Este mes is a time expression, and Spanish often places time phrases at the beginning to set the context: Este mes el alquiler sube...
You can move it, and the meaning stays basically the same:

• El alquiler sube un poco este mes. (very natural)
• Este mes sube un poco el alquiler. (also natural; slightly more emphasis on the increase)

In Spain, el alquiler commonly means rent (the money you pay monthly) and also the rental (the renting arrangement), depending on context.
In this sentence, it clearly means the rent payment.
(You may also hear la renta in some contexts, but el alquiler is the most everyday, Spain-neutral choice.)

• sube is the 3rd person singular present of subir: (él/ella) sube = it goes up.
• subir is the infinitive (to go up), so you wouldn’t use it here without another verb (e.g., va a subir).
• You don’t need a reflexive form. El alquiler sube is standard for “rent goes up.”

Yes—subir is very commonly used for prices, fees, rent, taxes, temperatures, etc.
So El alquiler sube = Rent goes up / increases.
Similarly:

• La gasolina sube. (Gas goes up in price.)
• Los precios suben. (Prices go up.)

un poco means a little. It functions as an adverbial phrase modifying sube.
Common placements:

• El alquiler sube un poco. (most neutral)
• El alquiler sube un poco este mes.
• El alquiler sube este mes un poco. (possible, but less common)

It’s very common (and often recommended) to use a comma before así que when it introduces a result/consequence clause:

• ..., así que debo ahorrar más. = ..., so I must save more.
  In informal writing, people sometimes omit the comma, but the comma helps readability.

así que means so / therefore / as a result and explicitly links cause → consequence.
entonces often means then, and can also mean so in conversation, but it can feel more sequential or conversational.
In this sentence, así que is a very clear “result” connector.

In modern Spanish (including Spain), deber + infinitive usually expresses obligation/necessity:

• Debo ahorrar más. = I must / I should save more.
  deber de + infinitive can express probability/assumption (more traditional/variable in use):
• Debe de estar en casa. = He’s probably at home.
  So here debo ahorrar is the correct structure for obligation.

debo can range from should to must, depending on context and tone. In everyday speech, Spaniards often use it similarly to should / I need to.
If you want a more “external obligation” feel, tengo que is very common:

• Tengo que ahorrar más. (I have to save more.)
  Your sentence with debo sounds like a personal conclusion/necessity.

Spanish usually omits subject pronouns because the verb ending already shows who it is:

• debo = I must
  You can add yo for emphasis or contrast:
• ..., así que yo debo ahorrar más. (emphasizes I, e.g., not someone else)

más here modifies the action ahorrar (“to save”), so it naturally comes after the infinitive:

• ahorrar más = to save more
  más ahorrar is not natural in Spanish in this meaning.

A few very natural alternatives:

• Este mes ha subido un poco el alquiler, así que tengo que ahorrar más. (present perfect; common in Spain for “this month it has gone up”)
• Este mes me han subido un poco el alquiler, así que debo ahorrar más. (someone raised it on me; common way to express the landlord increased it)
• Este mes el alquiler sube un poco, así que voy a ahorrar más. (intention: I’m going to save more)