In the 1891 census both James and William are with their uncle:
[Ancestry index]
Name: William Starkey
Age: 23
Estimated Birth Year: abt 1868
Relation: Nephew
Gender: Male
Where born: Isle of Man, Isle of Man
Civil parish: Marown
Ecclesiastical parish: Marown
[At: Stuggadoo, Marown]
Household Members: Name Age
Jacob Lewin 39
Ann Lewin 37
Jacob Lewin 13
Annie Lewin 11
Eleanor Lewin 7
James Starkey 26 Leadminer
William Starkey 23 Leadminer
[All birthplaces only Isle of Man.]

In 1901 James is still there, but William has gone (luckily).
Name: James Starkey
Age in 1901: 30
Estimated Birth Year: abt 1871
Gender: Male
Where born: Marown, Isle of Man, Isle of Man
Civil parish: Marown
Ecclesiastical parish: Marown
County/Island: Isle of Man
Household schedule number: 22
Household Members: Name Age
Jacob Lewin 45 Stone Quarry Man b.Braddan
Ann Lewin 44 b.Marown
Annie Lewin 20 b.Marown
James Starkey 30 Lead miner b.Marown

There was a Wm Starkie passenger from Liverpool in the time frame, but going to the USA:
passenger transcript details
Name: Wm STARKIE
Date of departure: 13 January 1892
Port of departure: Liverpool
Passenger destination port: Philadelphia, USA
Passenger destination: Philadelphia, USA
Date of Birth: 1866 (calculated from age)
Age: 26
Marital status: Single
Sex: Male
Occupation: Lab
Passenger recorded on: Page 1 of 5
Ship: LORD CLIVE
Official Number:
Master's name: Urquhart
Steamship Line:
Where bound: Philadelphia, USA
Square feet: 3728
Registered tonnage: 2206
Passengers on voyage: 147

[No place of origin given.]

Sue