Older Raypak heaters (more than 20yrs) had a cast or more of a brick like composition they called fire brick, and could or will tolerate water better than the newer fiber like material that they now call "refractory". I have seen this stuff after an exchanger has been leaking on it, and it doesn't do well being exposed to water. Now granted, washing the exchanger with water is not constant day after day exposure to water, and if you were to run the heater to dry out the material after the washing, who knows, maybe you will be ok. My opinion? would be to remove the exchanger to clean it. This is how we clean them, there is no water involved, and we can verify visually that the underside of the exchanger has gotten clean. Removal is not brain surgery, but i can see how it could be a bit overwhelming for the first time.
Now, having said that, some things to ponder as you move forward...
Exchangers soot UP, not OUT. There is nothing terminal about an exchanger that has sooted up, unless of course it has not been addressed for an extended period of time. Even then, the heaters protection switches can save the exchanger from running in that condition (heat). Also, 13 years is not necessarily terminal either. It all depends on the condition of the heater.
Another thing to address is how the heater got this way. Heaters don't soot up from age. They soot up from incomplete combustion or improper combustion (wrong air/fuel mixture). Sooted heaters generally suggest not enough air (indoors), or not enough air getting to them (high wind, or high altitude). If this heater is indoors, you should check for appropriate openings to the outside where the heater can draw air from. These openings have dimensions that need to be met. IOW, you can have an opening, but not a large enough opening. If outdoors, check for an equipment enclosure that while it has an open roof or ceiling, has four walls that are sealed at ground level by perhaps dirt, and extend a couple of feet above the heater.
And as Swampwoman stated, this all may be a mute point as you are (for now) the tenant, and really should not touch the heater other than to set the desired temperature. Perhaps, the best you could hope for is to get the landlord to pay for service, or if you were to get permission to have the heater serviced, and pay yourself.