What is the best approach to calculate long-term trend of runoff to precipitation ratio and \or evapotranspiration of a sub-basin when there is not any streamflow data available for that subbasin.
That is not good at all because there must be historic data for you to calibrate and validate your model. Any way, there are some ways and models that you can apply them. The easiest way is rational method. You can use this method as a bench mark to compare other methods with this, But bear in your mind that this model usually over estimates the run off. One of the best model that you can use in this case is Hec 1 method in Hec-hms model. That is the most reliable model. The only vagueness that I have in your case is that whether your basin is urban basin or not. If it is urban basin you can use EPA-SWMM model, but again bear in your mind that this model needs some parameters which their estimations is not very simple and you need to be very cautious while you are using this model. This paper is really helpful in this case:
"Conversion of Natural Watershed to Kinematic Wave Cascading Plane"
There are some other methods such as: unit hydrograph, SCS method and etc, but if you are looking for a model, Hec is highly recommended. I hope this would be helpful for you.
I suggest to interpolate precipitation and runoff fields and then to calculate the ratio using that maps. I mean you need to build the maps using information from surounding basins and then extract, as an aproximation, the ratio from data averaged for the basin of your interest. You can perform this with long term maps, but also for maps with annual resolution. First you need to study wich interpolation methods has the best performance for your study area, ussually the best results can be reached with optimal Gandin interpolation, Inverse distance method or Krigging interpolation. In the annual level you need to considere if there are any storage capacities that can influence the runoff coeffiecient calculation from one year to another. The rational method can be of help if you are working with a small basin.
If monthly temperature and precipitation data are available for the site you can use a monthly water balance model to estimate runoff and then compute a runoff ratio using mean annual estimated runoff to mean annual precipitation.
It seems that you get the discharge time series for a larger basin.
You can try first to estimate the specific discharge, ie the discharge/surface. Then you apply this ratio to the surface of the sub basin.
Second possibility. You then need to calculate the precipitations for the large basin and for the sub basin. Then you calculate the ration between discharge and rainfall for the large basin, and you apply it to the surface of the sub basin.