In that general area of the county, for the rent range you've specified, I would suggest also looking in nearby areas of Mount Vernon and New Rochelle. All of these areas have some parts that are not as desirable as others, due to the transitional nature of their current states of affairs. So, you might have to look in more places to find an apartment that you like.
You can easily link to train stations in New Rochelle and Pelham to get to NYC via local bus routes, the same as Yonkers. That said, you can also take some Westchester Bee Line buses to NYC subways in the Bronx, which allows MetroCard transfers and is less expensive on a monthly pass basis, but takes a bit longer than suburban MetroNorth train service.
Generally, if you're close to the Bronx in Yonkers, it's the Riverdale-VanCortlandt-Woodlawn areas and they're actually quite decent areas overall. You might also find an apartment in your range in this section of the city, but it would be a hunt as they are among the more expensive of Bronx offerings, especially Riverdale.
MetroNorth/MTA NYC subway/bus information:
MTA Home Page
Westchester Bee Line Bus information:
Bee Line Bus