Redirecting from one nginx to another

Since they're both using the same server_name, you can't do a simple redirection.

You can probably do some sort of proxy, though, along the lines of the following on the old server:

upstream newserver {
  server 172.16.0.1:80;  # this is new server, by IP address
}

server {
  listen 80;
  server_name subdomain.site.ru;
  location / {
    proxy_set_header Host $host;
    proxy_pass http://newserver;
  }
}

So, basically, configure the old server so that it will pass all requests to the new server. Of course, put in whatever configuration you'll need for client_max_body_size and all that.