Death certifcates were not required by the state of Pennsylvania until 1906. However, since you posted this query on the Allegheny County board, I'm assuming there may be a Pittsburgh connection, and the city of Pittsburgh required death registration beginning in 1870. If they died elsewhere in the county, there will only be a record if they had an estate administered or probated. Records for both of these can be obtained from the Register of Wills:
Allegheny County Register of Wills
City-County Bldg., 1st Flr.
414 Grant St.
Pittsburgh, PA 15219
412-350-4180
www.county.allegheny.pa.us/regwills/cert.asp

For naturalization records, there is a multi-volume book that lists all naturalizations in Allegheny County - it's available in many genealogical libraries, and the data from this index is also available through Ancestry.com, so if your ancestor was naturalized in Allegheny County, you should be able to find the info on Ancestry.com. The petitions themselves are available on microfilm through the National Archives if they went through a federal court. You could check whether their office nearest you has the film for western Pennsylvania, but it's unlikely. However, I think you can order a copy of the record through the mail - check with the National Archives to see what is available. But the first step would be to find your ancestor in the index and confirm what court the naturalization took place in.

Finding the actual arrival date would depend in finding the ship manifest that contains your ancestor's name. Many of these arrival records are also indexed on Ancestry, but it depends on which port they came through.