Installing a washing machine - sink drain pipe closed
Dishwasher Inlet

First of all, I assume we are talking here about a dishwasher and not a machine for washing clothing. It is unusual to connect a "washing machine" (as opposed to a "dishwasher") under the kitchen sink, though not impossible. I believe this is a difference in terminology between the USA and other parts of the world. Continuing based on dishwasher.


In the USA, many (most?) people have a disposal under the kitchen sink. I understand that is not nearly as common in other parts of the world. Most disposals (all the ones I've worked with, but I am not a professional plumber so my experience is limited) have a dishwasher inlet that looks very similar to the capped pipe. I believe the capped pipe is a dishwasher inlet. You should be able to remove (cut off if necessary, as close to the top as possible) the cap and attach the dishwasher hose directly to the pipe. The ridges on the pipe help to keep the dishwasher hose attached, though you can put on a clamp if it does not seem to be secure enough. If the dishwasher hose and the pipe are not compatible (the dishwasher hose should be just large enough to fit snugly over the pipe) then you will need an adapter - measure the hose and the pipe and go to your favorite hardware store.
When you install the dishwasher hose, it can be connected in one of two ways:


Air gap - a metal/plastic gadget that is typically installed next to the faucet. One hose goes from the dishwasher to the air gap. Another hose goes from the air gap down to the pipes. These are a pain (they often break over time, leaking around &amp; under the sink) but in some places they are still the only legal way to install a dishwasher.
High hose loop - the dishwasher hose loops up to just under the counter and then down to the pipes. This has the same general effect as a traditional air gap - preventing sewer water from backing up into the dishwasher - but is much simpler and easier to install.

As far as asking permission: If the landlord already knows you are installing a dishwasher then there should not be an issue. If the landlord does NOT know you are installing a dishwasher, then I recommend checking with him first. Cutting the cap off the pipe is no big deal, but installing a dishwasher - which can affect water usage and which (typically, unless you are using a portable unit, in which case cutting the pipes would not even be an issue) normally involves replacing a cabinet with the dishwasher - is something the landlord would expect to be consulted before installation.
While you're at it, the pipes look a bit strange. As already noted in comments, the trap is not a typical P-trap, though logically it should work. In addition, the pipe coming out of the trap is a flexible corrugated hose. That is generally not recommended as it can collect grease &amp; debris in the folds and clog more easily than a regular pipe. (You may say "but isn't the dishwasher hose the same?" and the answer is "the dishwasher doesn't get as much as the main drain pipe and the dishwasher hose is easily removable to clean if it gets  clogged".)