Irule or policy to URL rewrite

okay. You can use below

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"
		  HTTP::uri /[join [lreplace [split [string trimleft [HTTP::uri] /] /] 0 0 "down"] /]
		  pool Pool_other.u.test.com_1443
	      return
           }
	  }
       default {
	    return  
      }
    }
  }