Irule or policy to URL rewrite

There is still confusion around, how uri after /up should be translated. If you can provide more details on it then it would be clear. But I'm assuming those shouldn't be changed. Considering it won't change, you can use below iRule

when HTTP_REQUEST {
    set uri [string tolower [HTTP::uri]]
    switch -glob [string tolower [HTTP::host]] {
        "client.u.test.com" {
          if {$uri starts_with "/up" } {
		  HTTP::header replace "Host" "other.u.test.com"
		  pool Pool_other.u.test.com_1443
	      return
          }
	    }
       default {
	    return  
      }
    }
  }

If you need any other rewrite rule with uri after /up let me know and we can modify the iRule above