Russia to Ukraine...

[Greg G]

An alternative would be Mongolia. Here is an interesting photography report from last year, there is a hide, too:
Wild Saiga Antelope Watching and Photography: May 2019 - Mammal Watching
I think mammalwatching and cluodbirders would give more reports.

The same friend has been to Mongolia too, with a local tour operator. He was very focused on photographing particular birds, but saw wild Przewalski horses and many other mammals.

From his description, Mongolia is also well organized tour, and again there are local tour operators, but one needs a week or two due to long distances. However there is lots of other steppe wildlife besides saiga during this time. And it is possible doing it on moderate prices, much lower than e.g. a similar time in famous African national parks would cost.

The Mongolia saiga is the rarer subspecies of the two, and is actually very rare and even more endangered than the nominate subspecies. It also looks slightly different. You should be very lucky to have good pictures of them on a short trip. I have no idea about tours. I have spoken to Mongolian saiga experts, they were very nice and knowledgeable, but I know nothing about tours. If you are happy with birds, wild horses and wild asses, you will not be disappointed I guess.