Nomenclature of alkoxy and halo groups
Your name, 1-chloro-2-methoxyethane, is correct.


Here is a link to the relevant page of the 1993 edition of the IUPAC Blue Book - Nomenclature of Organic Chemistry, which is published online by ACD Labs with permission of IUPAC.

Here are some example ethers and sulfides from that page, and we see a similar case to yours:



There is no difference in preference or priority between halo and alkoxy substituents (or any of the other prefix only substituents). Thus, in a case where there are two such groups and assignment of locants generates the same combination either way, you choose to assign the lower locant number to the alphabetically first.