On one photo, you see that there is a large sink, which you will use to clean. Note: you have only cold water, and in mountains it could be really cold.
You can clean up yourself pretty well, with standard shower gels. It is just slower to get the gel out of you (but remember cold water, so maybe it is a good thing).
I think for few days it is ok. For more than a week, look for some alternate places to do some real showers with hot water (e.g. in a public hut, or on the valley village, e.g. in a sport center or swimming pool).
Normally the shower, bad, and toilets are like in the rest of western Europe. In fact one of the largest producer of toilets is Swiss (you will find the name on all continent). It is just that in huts (so not for living all year around), you may not have hot water, sewage system, and freezing winter (so danger of freezing water), so the system is like on older times. But on most of public huts (but not the most remotes) you will have hot water, showers (solar or gas boilers). Also on many private houses you get some more services.
Your hut seems something more like an adventure site, e.g. for schools, few days. To me it seems strange to have so few items on a huts which is connected with roads (gas boiler is very cheap).