There seem to be 2 thing at play here.
The first is tuning. You've already been given suggestion on how to intonate the instrument but one thing that hasn't been mentioned is the hight of the slot in the nut. If it's cut too high, like it often happens on cheaper instrument, it might make it difficult tuning/intonating the instrument properly: you might get an open string and twelfth fret in tune while the first fret will be sharp, and then getting progressively flater as you move to higher fret; or tune the first fretted not and twelfth then the open string will be off. The cure for that is simple, just get a nut slotting file and cut the groove deeper (it will be easier to play too as a bonus).
The other is tone. Well, that is to be expected; open strings will sound different than fretted ones, and moreso with cheaper instrument since cheap plastic is used for nut material. The nut could be replaced with better material, but it will never sound the same as a fretted note even then.