TL;DR: The final product will depend on what solvent you are using, what temperature the reaction is being operated and the concentration of the reactants.
Long answer
From the abstract of this paper1:
The $\ce{AlCl3}$ and $\ce{SnCl4}$- catalyzed Friedel Craft alkylation (methylation, ethylation, isopropylation and tert-butylation) of naphthalenes with alkyl halide was studied in nitromethane, carbon disulfide and benzene solution. Alkylation in nitromethane shows least isomerization giving 75-100% α and 25-0% β substitution. Ready isomerization of αalkylnaphthalene under usual Friedel-Crafts conditions leads to substantially increased β-alkylnaphthalene formation. To ascertain the effect of acid-catalyzed isomerization, the $\ce{AlCl3}$ catalyzed isomerization of α and β-alkyl (methyl, ethyl, isopropyl and tert-buty) naphthalenes were studied, including the determination of equilibrium composition of α/β isomer pairs.
Similar thing happens for FC-acetylation2,3:
The Friedel–Crafts acetylation of naphthalene has been shown to give varying α : β isomer ratios as a function of concentration of the reactants. In consequence the ratio is strongly time- and temperature-dependent. Values of the α : β ratio (determined by an extrapolation procedure, for 1,2-dichloroethane solution) for the limit of zero reaction time, were plotted against reactant concentration, to give a straight line having a near-zero intercept. From this it is inferred that, whereas β-acetylation is essentially first-order with respect to the acylating reagent, α-acetylation is dominantly second-order in acylating reagent.
The Friedel–Crafts acetylation of naphthalene, using acetyl chloride and aluminium chloride in 1,2-dichloroethane solution, has been studied kinetically. The α/β isomer ratio changes as a function of concentration of reactants, and time, e.g. from an initial 4–5 to a final 0.7.

(full image here)
Notes and References
- Aromatic substitution. XXXVII. Stannic and aluminum chloride catalyzed Friedel-Crafts alkylation of naphthalene with alkyl halides. Differentiation of kinetically and thermodynamically controlled product compositions, and the isomerization of alkylnaphthalenes, George A. Olah and Judith A. Olah
Journal of the American Chemical Society 1976 98 (7), 1839-1842
DOI: 10.1021/ja00423a032
- Kamounah, F. S., Andreou, A. D., Bulbulian, R. V., Gore, P. H., Miri, A. Y., & Waters, D. N. (1981). The Friedel-Crafts acetylation of naphthalene. Evidence for concurrent second- and third-order reactions. Journal of the Chemical Society. Perkin Transactions 2 (2001), 2, 376-378. [55]. https://doi.org/10.1039/P29810000376
- The Friedel–Crafts acetylation of naphthalene in 1,2-dichloroethane solution. Kinetics and mechanism, David Dowdy, Peter H. Gore and David N. Waters, J. Chem. Soc., Perkin Trans. 2, 1991, 1149-1159 , DOI: 10.1039/P29910001149
- The acylation of naphthalene by the friedel–crafts reaction, J. Chem. Soc., 1949, S99-S103, DOI: 10.1039/JR9490000S99 (acetylation of naphthalene using methylene and ethylene chloride. In absence of solvents, α-substitution occurred and in presence of solvents at higher temperture, β-substitution occurred)
- FRIEDEL-CRAFTS ACETYLATION OF 1- AND 2-METHYLNAPHTHALENE By HUBERT EMMANUEL, 1978 (link)