When you calculate your P_{1} = x_{1}*P, you are not solving for P_{Ne}, you are solving for P, so when you get the answer 0.2785, that is your total pressure not the partial pressure of Ne, remember they gave that to you (8.87kPa or 0.08754 atm).

So, you already have the partial pressure of Ne. You know the formula, P_{1} = x_{1}*P, solve that for P. Don't go to all the trouble of calculating the ratio of Ar+CH_{4} to Ne and such. Also, you already know the ratio (you calculated the mole fractions). For future reference, you just need the mole fraction of one gas and it's partial pressure to find the total pressure. After that your golden.