Learn Hardcore Norwegian: Jeg må håndtere budsjettet vårt bedre i år. - I have to handle our budget better this year.

Må is a modal verb meaning must / have to. It expresses necessity or obligation (often practical necessity): I have to manage our budget better this year.
If you used bør, it would sound more like advice (should). If you used skal, it can mean intention/plan (will / am going to) or obligation depending on context.

Why is the verb after må in the infinitive form håndtere?

After Norwegian modal verbs like må, kan, vil, skal, bør, the next verb is typically in the bare infinitive (infinitive without å).
So you say Jeg må håndtere ..., not Jeg må å håndtere ....

Can I also say Jeg må å håndtere budsjettet vårt ...?

No—standard Norwegian does not use å after a modal verb.
Å is used in many other infinitive contexts, e.g. Jeg prøver å håndtere budsjettet (I try to manage the budget).

Why is it budsjettet and not et budsjett?

Budsjettet is the definite form: the budget. In this context, you’re talking about a specific known budget (your household/company budget), so definite is natural.
Et budsjett would be a budget, which sounds like you mean any budget or you’re introducing it for the first time.

What’s going on with vårt—why is the possessive placed after the noun?

Norwegian commonly places the possessive after the noun: budsjettet vårt = our budget (literally the budget our). This is very common in everyday Norwegian and often sounds more natural than the “English-like” order.

Can I also say vårt budsjett? Does it change the meaning?

Yes, vårt budsjett is correct too.
A common guideline:

budsjettet vårt is neutral/common, often “the budget we already have”
vårt budsjett can sound a bit more emphatic/contrastive: our budget (not someone else’s), or slightly more formal depending on context

Why is it vårt and not vår?

Because budsjett is a neuter noun (et budsjett). Possessives agree with gender/number:

vår = common gender singular (en-nouns)
vårt = neuter singular (et-nouns)
våre = plural
So: budsjettet vårt, but bilen vår, and planene våre.

Where does bedre go in the sentence, and could it be placed elsewhere?

Bedre is an adverb modifying the verb phrase håndtere budsjettet. The placement after the object is very natural:
Jeg må håndtere budsjettet vårt bedre i år.
You could also move bedre earlier for emphasis, but it can sound marked:

Jeg må bedre håndtere budsjettet vårt i år. (possible, but more formal/less common)

Why is i år at the end? Could it go earlier?

Time expressions often go late in Norwegian sentences, especially in neutral statements. End position is very common: ... bedre i år.
You can move it for emphasis:

I år må jeg håndtere budsjettet vårt bedre. (focus on this year)
If something other than the subject comes first, Norwegian uses V2 word order (see next question).

If I start with I år, why does it become I år må jeg ... and not I år jeg må ...?

Norwegian has V2 word order in main clauses: the finite verb (here må) must be the second element in the sentence.
So when I år is moved to the front (element 1), må must come next (element 2):
I år må jeg håndtere budsjettet vårt bedre.

How do I pronounce tricky parts like håndtere and budsjettet?

General tips (exact pronunciation varies by dialect):

håndtere: the å is like the vowel in British law; nd is usually pronounced (often a bit “compressed”). Stress is typically on the second syllable: hån-TE-re.
budsjettet: looks long, but it’s said smoothly. The dj is like a y/j sound in many dialects (similar to the s sound in English measure in some speakers, but often more like sh/y). Stress is usually on the last part: bud-SJE-ttet.

Is håndtere the most common verb here, or could I use something else?

Håndtere is correct and fairly common, meaning handle/manage (often “deal with” in a practical way). Depending on context, you could also use:

styre: very common for budgets, meaning manage/steer
- Jeg må styre budsjettet vårt bedre i år.
forvalte: more formal, like administer/manage responsibly
Choose based on tone: styre is very everyday; forvalte is more official.

How would I negate this sentence?

Put ikke after the finite verb (må) in a main clause:

Jeg må ikke håndtere budsjettet vårt bedre i år.
Be careful: this often means I must not / I’m not allowed to, not “I don’t have to.”

To say I don’t have to, you’d typically say:

Jeg trenger ikke å håndtere budsjettet vårt bedre i år.
or
Jeg behøver ikke å håndtere budsjettet vårt bedre i år.