Which carbocation is more stable, the ethyl- or 1-propyl-carbocation?
The ethyl carbocation ($\ce{C2H5+}$) has a different structure than what you might have anticipated.  It is a non-classical ion with a bridging hydrogen.
For a description of just what a non-classical carbocation is see this answer. 
The hallmark of a non-classical ion is 3 atoms with 2 electrons spread over them.  This is called a 3-center 2-electron bond (hypercoordinate bonding). In the case at hand the 3-atom 2-electron bond involves the 2 carbon atoms and the bridged hydrogen atom.



So there is no "methyl" group present in the ethyl carbocation that can inductively release electrons.  Additionally, the hyperconjugated resonance structure you mentioned has actually been replaced by a hydrogen that is partially bonded to the other carbon atom.

The n-propyl carbocation would likely have a similar non-classical structure - if it existed at all.  This carbocation has not been experimentally observed, it rearranges too rapidly to the much more stable 2-propyl carbocation.

So while we can't answer your question about whether inductive or hyperconjugative effects are more important in this series of compounds, we can assess which cation is more stable.

A variety of experiments suggest that the heat of formation of the non-classical ethyl cation is around 215 kcal/mol (reference, see pp. 68-72). This same reference points out that the heat of formation of the classical 1-propyl cation lies approximately 20 kcal/mol above that of the 2-propyl cation.  The heat of formation of the 2-propyl cation is around 193 kcal/mol.  Adding 20 kcal/mol to that value would place the classical 1-propyl cation around 213 kcal/mol. But, assuming the 1-propyl cation would also be a non-classical ion it would be lower in energy than the classical version, its heat of formation would therefore be less than 213 kcal/mol.

This analysis suggests that the 1-propyl cation would likely have a lower heat of formation than the ethyl cation.  In other words, the 1-propyl cation would likely be more stable than the ethyl cation.