الفرن كان نضيف امبارح، بس دلوقتي محتاج يتنضف تاني.

كان is the past-tense form of to be.

• الفرن نضيف = The oven is clean
• الفرن كان نضيف = The oven was clean

In Egyptian Arabic, the present-tense is/am/are is usually not said, but in the past you normally use كان.

Because الفرن is a masculine singular noun, so the adjective must match it.

• الفرن → masculine singular
• نضيف → masculine singular adjective

If the noun were feminine, you would usually use نضيفة.

Also, نضيف is a common Egyptian colloquial form/spelling corresponding to MSA نظيف.

No. Arabic often does not need a separate subject pronoun when the subject is already clear.

In الفرن كان نضيف, the subject is already الفرن.

In محتاج يتنضف, the verb يتنضف already implies it through its form:

• يـ at the beginning marks he/it in the present tense

So the meaning is naturally understood as it needs to be cleaned.

امبارح means yesterday in Egyptian Arabic.

It works as a time expression by itself, so you do not need a preposition like in or on before it.

• كان نضيف امبارح = It was clean yesterday

This is similar to English, where you also just say yesterday, not in yesterday.

Here, بس means but.

So:

• الفرن كان نضيف امبارح، بس... = The oven was clean yesterday, but...

In other contexts, بس can also mean only or just, so its meaning depends on context.

دلوقتي means now or right now in Egyptian Arabic.

So:

• بس دلوقتي = but now

It is a very common Egyptian word. A common pronunciation is roughly dilwa'ti.

محتاج literally means needing or in need of, but in natural English it is often translated as needs.

So:

• الفرن محتاج يتنضف literally = The oven is needing to get cleaned naturally = The oven needs to be cleaned

Like adjectives, محتاج agrees with the noun:

• masculine: محتاج
• feminine: محتاجة

Since الفرن is masculine, محتاج is the correct form.

يتنضف means to get cleaned or to be cleaned.

It comes from the idea of cleaning, but in this form it has a passive/reflexive sense: not to clean something, but to become cleaned / be cleaned.

So:

• نضّف = to clean
• يتنضف = to get cleaned / be cleaned

That is why محتاج يتنضف means needs to be cleaned.

تاني here means again.

Arabic often places words like again after the verb phrase or at the end of the clause, so:

• محتاج يتنضف تاني = needs to be cleaned again

That word order is very natural in Egyptian Arabic.

Also, تاني can mean again, another, or second, depending on context.

Yes, but there is a difference in style/register.

• نضيف is very natural in Egyptian colloquial
• نظيف is closer to Modern Standard Arabic spelling/pronunciation

In everyday Egyptian writing, you may see either, but نضيف fits the colloquial sentence very well.

Yes, Arabic word order can vary, but starting with الفرن is very natural because it sets the topic first:

• الفرن كان نضيف امبارح = As for the oven, it was clean yesterday

This is a common way to build sentences in Arabic: mention the thing you are talking about first, then describe it.

Yes, very much so. Several words are strong clues:

• امبارح = Egyptian yesterday
• بس = common colloquial but
• دلوقتي = Egyptian now
• نضيف = colloquial form of clean
• يتنضف = colloquial verbal pattern
• تاني = colloquial again

A more formal MSA version would be something like:

الفرن كان نظيفًا أمس، لكن الآن يحتاج إلى أن يُنظَّف مرةً أخرى.

So the original sentence sounds natural and everyday in Egyptian Arabic.