I don't know if is correct to use "the month" with "starts".
Is correct to use "to" Store or is "at" better?
Here below the phrase:
The month of the Special Summer starts to Archivio Dondup!
Note: Archivio Dondup is the name of a store.
Yes, you can use the article the with month, but your sentence raises a lot of questions.
Is "Special Summer" the formal name of an event? (The capital letters make it appear that way, like it's the name of an established summer sale.)
If Special Summer is not a proper noun, I would suggest this:
(Yes, at is the correct preposition to use).
If Special Summer is a proper noun, then perhaps there are better ways to write this:
Also, in English, a season (such as summer) can be used adjectivally:
In this case, I prefer a to the, although using the would not be grammatically incorrect. (Using the instead of a would suggest that you expect the readers to be familiar with the summer specials at this store, either from past years, or else from a preceding advertising blitz.)
Given that you have a store name, the sentence would be:
Which would imply that the store is having a summer sale or a stock full of summer products. The preposition
Alternatively, you could use something like the following:
As @Ijacqu says, you wouldn't say "the summer starts to ..." "To" indicates direction or movement, and the summer isn't going anywhere. Also, "starts" doesn't take a direction.
It is fairly common to say that something "starts at ..." This is the first place where the event will happen or where it begins. You could mean this literally, like, "Summer starts at the international date line at midnight on the summer solstice." Or in this case, figuratively, you must go to this store as the first step in enjoying your summer.
You can also say "starts with ...", meaning that you need to do this thing or have this thing to start. Again, it could be literal, like "The engine starts with the press of the start button." Or figurative, "Summer starts with [our store]", meaning, our store is what you need to begin your summer, or where you go to begin your summer.
I'm not sure what you're trying to say with the word "month". You could say, "The month starts at [our store]". That's grammatically correct, but I don't know what idea it would be intended to convey. What month? What is significant about the beginning of the month? It certainly could make sense in a totally different context. Like, "At our company, the month starts with an inventory count."