Did Helsinki have higher standards of living than Saint Petersburg when it was a part of the Russian Empire?

Firstly, this greatly depends on how the GDP was back-calculated for those times. This is a difficult task as I've described elsewhere here.

For reference, I've copied in the table for a quick comparison between "Finland" and "Russia" between 1830 and 1913 as given on the OP's link under "Europe 1830–1938 (Bairoch)":

I quickly graphed the above for a more illustrative version:

If we look at this alone, it does look as if the people of Finland would have experienced a 'better' lifestyle (let's leave defining this out of the scope of this answer).

The 1897 census is useful in thinking through the above:

The total population of the Russian Empire was recorded to be 125,640,021 people (50.2% female, 49.8% male; urban 16,828,395, median age of 21.16 years).

Finnish population is not counted per se as the census did not take place there (to which we will return later), but it's population for 1897 is given as 2,610,300 people by Statistics Finland. Meanwhile, Helsinki had 66,734 people in 1894 which makes it approx. the 31st city by population in the Russian Empire.

With regards to Finland's development and economy, though Wikipedia laments its agrarian outlook, 'The Finnish economy 1860-1985 : Growth and structural change' is more illustrative:

Significant legislative reforms were carried out in Finland during the 1860s: a prohibitive order on the establishment of steam-powered sawmills was repealed (1857), Parliament was reconvened after an extended adjournment (1863), Finland gained its own monetary unit (1860-1865), trade guilds were abolished (1859, 1868), and so on. These reforms are regarded as being so important that many researchers have even dubbed the 1860s as the decade of Finland's Industrial Revolution, and the legislative reforms as the starting motor of growth.
_{—Hjerppe, 'The Finnish economy 1860-1985 : Growth and structural change'}

The result of these reforms is the growth in prosperity that the above GNP chart displayed in pretty good detail. Meanwhile, Russia proper really was rather agrarian in its outlook—or, well, to be precise, it had more industrialization which was ongoing but this was centralized around the European Russian hub towns (St. Petersburg, Moscow, Kiev, Riga, etc...) while large areas of the country remained agricultural.

However, this hasn't really got us much closer with respect to "Helsinki & St. Petersburg" as opposed to "Finland & Russia". I tried some specific searches, but came up quite empty so I'll go without numbers and only expand on my logic:

• Most importantly, Imperial Russian government could not have moved to an autonomous duchy;
• St. Petersburg (and Moscow) were 'Russian' cities and not 'Swedo-Finnish' ones as Helsingfors/Helsinki;
• That Finland was an autonomous grand duchy also meant that Finland had its own nobility who was generally loyal to the Emperor, so suggesting that the Russian nobility move into its 'territory' would have caused unrest amongst the locals;
• Skilled labourers in both cities would have been better off than unskilled ones;
• No matter where the nobility was, their income (from property, etc...) would have been sufficient to afford them whatever they wanted so they would not have had any interest in whether generic housing was affordable or what a skilled labourer would make;
• Food and housing in cities and towns would have been higher than in rural areas, but this effect would have been similar between Helsinki & surrounding areas and St. Petersburg & its surrounding areas;
• The smaller population of (the Swedish- & Finnish-speaking) Helsinki would have made it an unattractive residence for a primarily Russian-speaking nobility (who were obviously conversant in French, but...) who would not have found entertainment as easy;
• Family residences and townhouses were already established in St. Petersburg and the nobility would have required a very good reason to move, en masse, to Finland because of the large investments involved (but note that many had some residences in rural Finland, incl. the Emperor who had a summer fishing house near modern Kotka);
• St. Petersburg, a far larger city, would have had much more allure for the Russian nobility, as a place of residence for those same reasons;
• Wanderers and seasonal workers were likely to arrive in larger numbers in the bigger of the two cities (St. Petersburg) as there was more work for them there, but this is also likely to skew the numbers slightly downwards (more in St. Petersburg; less in Helsinki).