Direction from Shinano Omachi to Tateyama

Is it possible to take the Kurobe Alpine route in the direction from Shinano Omachi to Tateyama? Can we buy such tickets in advance? We would come in summer from Matsumoto.

Yes, you can travel along the route in both directions. You can also start and end at the same side, backtrack or travel in any other way you wish.

Can we buy such tickets in advance?

Advance reservations are possible only for the first sections of the route, as explained here: https://www.japan-guide.com/e/e7551.html

For other tickets, you buy them on the day.

Alternatively, the Alpine-Takayama-Matsumoto Area Tourist Pass covers the Alpine Route in which case you would need to purchase additional tickets:
https://www.japan-guide.com/e/e2361_alpen.html

You can buy Ogisawa to Tateyama section online.

https://tateyama-kurobe-webservice.jp/AlpenTour/html/VW001W0010.html

The ticket pickup points are Ogisawa and Tateyama (in your case Ogisawa). From Shinano-Omachi to Ogisawa you must buy bus ticket on-site at Shinano-Omachi. There is a ticket office for the bus. No reservation is required. I am not sure whether you can use credit card so prepare some cash (JP yen).