How can I improve my model on a very very small dataset?
From what you say, I think you should start with checking three options:

I) Ordinary least squares (OLS): Just run a „normal“ linear regression. This will not yield great predictions, but you could view the model as a causal one, if you can assume a linear relation between $y$ and $x$. When you have five predictors and 35 observations, you have a total of 29 degrees of freedom which is „okay“. When you estimate the model in „levels“, so just values as they are, you can directly interprete the estimated coefficients as marginal effects. E.g. a model $y=\beta_0+\beta_1 x + u$, tells you that when $x$ increases by one unit, $y$ changes by $\beta_1$ units, just like a linear function.

II) You can use Lasso/Ridge/Elastic Net: All of them are linear-like models with a penalty term to „shrink“ $x$ variables if they are „not useful“. This works like automatic feature selection if you like to say so. There is a great package by Hastie et al. for R. You can find it here. It is also available for Python. 

III) Maybe (!) boosting could be an option as well: You would (likely) need to do some feature selection/engineering on your own. But Boosting is able to work with a small number of observations, with highly correlated features, and it often works well with highly non-linear problems. There are LightGBM or Catboost as possible Python packages. Find some minimal examples here.

With II) and III) you will find that you are not really able to „set aside“ a number of observations to check if your models work (because you don’t have much data). You could use cross validation (Ch. 5 in ISL, link below), but you need to see how it works. Instead of going for a predictive model, I tend to say that you might be better off starting with a „causal-like“ OLS model. With OLS you do not really need a „test-set“. OLS is very robust. 

Since you seem to be new to statistical modeling, you might benefit from having a look at „Introduction to Statistical Learning“ (Chapters 3 and 6 in particular). The PDF is online and there is code for the Labs in Python and R. The advanced book would be „Elements of Statistical Learning“. 

Good luck with your project!