اليوم أحتاج إلى سوق كبير قريب، لكن ليس لدي وقت.

A careful MSA-style reading (with case endings) could be:

• al-yawma aḥtāju ʾilā sūqin kabīrin qarībin, lākin laysa ladayya waqtun. Stress is generally on a “heavy” syllable near the end of each word, e.g. -yawm-, -ḥtā-, -qī-, -bī-, -rī-, -lay-, -day-, -waq- (Arabic stress rules are fairly regular compared to English).

Yes. اليوم (today) is a time expression, and it’s very common to place time (or place) expressions at the beginning to set the scene/topic. You could also place it later, but starting with it feels natural and clear.

أحتاج is the imperfect/present form, 1st person singular: I need / I am in need (of).
It comes from the verb اِحْتاجَ (to need). In MSA, the imperfect often covers both present and general/near-future meaning depending on context.

In MSA, اِحْتاجَ typically goes with إلى: أحتاج إلى ... = I need ... (literally: I need to/toward ...).
Using it without إلى is not the standard MSA pattern.

In fully inflected MSA, one possible vocalization is:

• اليومَ أحتاجُ إلى سوقٍ كبيرٍ قريبٍ، لكن ليسَ لديَّ وقتٌ. Key points:
• اليومَ is often treated as a time adverb (ظرف زمان) and appears accusative (-a).
• After إلى, the noun is genitive: سوقٍ.
• Adjectives follow and match case/definiteness/gender: كبيرٍ قريبٍ.
• وقتٌ is the subject of the nominal clause and is nominative (-un).

(In real life, case endings are often not pronounced, especially in everyday speech.)

In Arabic, descriptive adjectives normally follow the noun:

• سوقٌ كبيرٌ = a big market
  Putting the adjective before the noun is not the normal “plain adjective” pattern in MSA.

Because سوق here is indefinite (a market), the adjectives are also indefinite: سوقٍ كبيرٍ قريبٍ.
If you make the noun definite, the adjectives must also be definite:

• إلى السوقِ الكبيرِ القريبِ = to the big nearby market / the nearby big market

Both orders are possible. Often, the more “important/basic” description comes first, and the more “situational” one comes after, but it’s flexible.

• سوق كبير قريب can feel like: a big market (that is) nearby.
• سوق قريب كبير can feel like: a nearby market (that is) big.

لكن means but and introduces a contrast. A comma is just punctuation style; it’s common to separate the clauses in writing:

• ..., لكن ... = ..., but ...

MSA typically does not use a single verb “to have” the way English does. Possession is usually expressed with a prepositional structure:

• لدي / عندي = I have (literally: with me / at me)
  So ليس لدي وقت is literally: There is not, with me, time → I don’t have time.

لدي is لدى (with/at) + the suffix ـي (my / me).
Many texts write it as لديّ with a shadda on ي to show pronunciation (ladayya) and to avoid confusion, but both spellings are seen.

Yes. ليس عندي وقت is also correct and very common.
A rough guideline:

• لدي is often felt as a bit more formal / more “MSA”.
• عندي is extremely common and can feel a bit more everyday, though it’s still fine in MSA.

Both mean I don’t have time here.