I have tried a similar method for p-methoxyphenol
Substances: 1 mol p-methoxyphenol, 1.1 mol DMS, 800 mL 20% aq NaOH, 2 mL acetone per gram phenol.
Method: 4-Methoxyphenol was introduced into a 3 L erlenmeyer flask equipped with a stir-bar and dissolved in acetone. The mixture was gently stirred (yellowish colour). The aqueous NaOH and DMS were added in portions: 75-100 mL aq NaOH + 15-20 mL DMS. The reaction was rather exothermic. When all DMS was added, the reaction mixture was gently heated (35-40°C) overnight (ca 6 hours). Acidification of the reaction mixture (concentrated HCl) and extraction with ether gave a dark burgundy mixture, which was washed with aq NaOH till the aqueous phase became colourless. Removing the solvent under reduced pressure yielded 133.4 g of a yellow oil that crystallized upon standing. GC is pretty OK (97%).
Purification via distillation may be necessary, depending on the use of the p-dimethoxybenzene.
Substituting DMS for MeI: buy PiHKAL. It's full with MeI methylation examples. Or UTFSE. Plenty of examples. However, ring methylation side reactions might occurr.
Acetone: the use of acetone as co-solvent helps preventing the saponification side-reaction of DMS, hereby increasing the yield and the amount of reagent. Substituting DMS for DES has been proven to work as well, though yields are about 5% lower.