Outdoor wiring is the way to go for this case. Try this cable : AMP Netconnect [1859222-1714]<-- White Cable if your outdoor wall is white. Full Specs
Link.
It is, "XG CAT 6A" grade cable I use this for most of my projects that includes outdoor to riser wiring and 10G Copper Wiring. The newer model is [1499389] this one is the EU ROHS compliant.
Indoor wiring also possible but it will increase cost but negligible for this case. Maybe 10 to 20 more $$$. You can use the same wire as above or, use the cheaper Unshielded CAT6. and route it from your stairs.
If you want to avoid hassle, wireless is the way to go. But, expect lower performance than cable. Wireless performance can be on par with cable only if, you willing to spend as much or rather more money compared to cable + cabling [
for this case only]. Those equipments are usually use for wireless link above 5km or penetrating 5 storey of 250mm suspended slabs with minimal performance degradation.
If want to go extreme, try fiber optics using SFP or transceiver.
Another alternatives, try MoCA. Ethernet over Coaxial a.k.a Starhub SCV point.