The Swedish post road from Norway, through Sweden, used the Åland archipelago to pass into Sweden, and this is easily found (evidence of) in the south of Finland to the present day. When (and where) was the first overland route constructed overland from Sweden into (Swedish) Finland?
The only (poor) evidence I have for roads existing in the north is by the War of 1808–9 where Russian forces were planning to advance overland into Sweden (along with an army group advancing across the Gulf of Bothnia). One of the WP article's references does say "In addition, several new good roads had been built into Finland greatly reducing the earlier dependency on naval support for any large operation in Finland." but it doesn't specify where these roads were.
I looked through all articles on Swedish and Finnish road networks on the English Wikipedia, and the most I found was a reference to a 'Finnmark path' which was meant to have gone from Finnish Lapland to Finnmark in the 16th century. The Finnish WP article for the same page does not mention the Finnmark path at all, and I couldn't find anything else on a road of that name.
I understand—from the comments—that the term "road" can be meaningless without further definition for a period much longer than a few centuries ago. For clarity, I'm defining road as purpose-built (or purpose-developed) and used regionally for that purpose, such as the post road mentioned above. This would mean hunting tracks that slowly developed don't count, while a merchant-led endeavour to expand (and maintain) the tracks between two townships would.