Sábado vou correr no parque.

In Portuguese it’s common, especially in spoken or informal contexts, to drop em or no when the day of the week starts the sentence.
Equivalent to “On Saturday, I’m going to run in the park,” but more concise.
You can also say No sábado vou correr no parque for a slightly more formal or explicit version.

You’re right: in Portuguese, days of the week are normally lowercase (segunda-feira, terça, sábado, etc.).
Here Sábado is capitalized simply because it’s the first word of the sentence. Otherwise it would be sábado.

Portuguese is a pro-drop language: the verb ending -o in vou already shows first-person singular.
Adding eu is grammatically correct (Eu vou correr), but speakers often omit it unless they need emphasis.

Vou correr is the periphrastic future, formed with the present of ir + infinitive. It expresses a planned or near-future action (“I’m going to run”).
The simple future (correrei) exists but is more formal and less common in everyday speech.

Portuguese standardly uses ir + infinitive directly (vou correr).
The equivalent ir a + infinitive is either literary/archaic or regional for continuous actions, but it’s not used as the default future marker.
For most speakers and contexts, vou correr is the natural choice.

1. parque is masculine singular, so the article is o.
2. You contract em
   • o = no.
3. You need the definite article with location nouns:
   • no parque (correct)
   • na parque would imply a feminine noun (there is none)
   • em parque (without article) is ungrammatical in this context.

Add a qualifier:
• No próximo sábado vou correr no parque (“Next Saturday…”)
• Or more colloquially: No sábado que vem vou correr no parque.
Without such qualifiers, Sábado in context usually means the coming Saturday.