Ilmoitustaululla näkyy uusi ilmoitus vesikatkosta.

The subject is uusi ilmoitus (a new notice). Finnish typically marks the subject by case rather than position, and here uusi ilmoitus is in the nominative singular.
Because the subject is singular, the verb is also singular: näkyy = (it) is visible / can be seen.

Ilmoitustaululla is the adessive case (ending -lla/-llä), which often answers where? in the sense of on / at something.

• ilmoitustaulu = noticeboard
• ilmoitustaululla = on the noticeboard / at the noticeboard

So it functions as a location adverbial: the place where the notice is visible.

Finnish word order is fairly flexible. Putting Ilmoitustaululla first is common when you want to set the scene (topic-first): as for the noticeboard, there is…
You could also say:

• Uusi ilmoitus vesikatkosta näkyy ilmoitustaululla.
  This shifts emphasis toward the new notice rather than the location.

näkyy is the 3rd person singular present of näkyä = to be visible / to be seen. It highlights that the notice is something you can physically see (e.g., posted on the board).
A very natural alternative is:

• Ilmoitustaululla on uusi ilmoitus vesikatkosta. = There is a new notice … on the noticeboard.
  Using näkyy adds a nuance like it’s (clearly) visible there.

Finnish doesn’t have a/the, so it’s inferred from context. In a neutral statement like this, uusi ilmoitus usually reads as a new notice (new information appearing). If it’s a specific known notice, context (or extra words like se = that/it) clarifies it.

vesikatkosta is the elative case (-sta/-stä), which literally means out of/from, but it’s also used to mean about / concerning with communication nouns/verbs.
So ilmoitus vesikatkosta = a notice about a water outage.

Other possibilities exist depending on nuance:

• ilmoitus vesikatkosta = about the outage (topic)
• ilmoitus vesikatkosta is generally more “about/concerning” than a plain “of” relationship.

Yes, sometimes you will see ilmoitus vesikatkosta and ilmoitus vesikatkosta is very common for about.
Using the genitive vesikatkon would be more like a notice of the water outage (a bit more like naming/announcing the outage itself). In real usage, both can occur; -sta is a very standard way to express the topic.

It’s a compound noun:

• vesi = water
• katko = interruption/cut (from katkaista = to cut/break off)

So vesikatko means water outage / interruption in water supply.

Yes. Adjectives typically agree with the noun in number and case. Here both are nominative singular:

• uusi (nom sg) + ilmoitus (nom sg)

If the noun changes case, the adjective usually changes with it, e.g.:

• uudesta ilmoituksesta = about/from a new notice (both in elative)

Not in a way that changes the visible consonants here. These forms are straightforward:

• ilmoitustaulu + -lla → ilmoitustaululla
• vesikatko + -sta → vesikatkosta

What you do see is typical Finnish case attachment and long compound words, but no tricky gradation like k → v etc. in these particular forms.