Cheers. NP
There aren't a great deal of museums around that area. But If you're interested, I'd recommend heading down to the Fine Art institute just off the way down Gran Via. Get up on the roof terrace or hobnob in the bar. Spectacular.
https://www.circulobellasartes.com/azotea/
The 3 big must see museums are all down on the Paseo Prado which is kind of the other way. This is the route to see all of them, plus the above. As you can see the centre of Madrid isn't big. About 1/2 an hour each way from Sol takes you to the edge.
https://www.google.com/maps/dir/Museo+Nacional+Centro+de+Arte+Reina+Sofía,+Calle+de+Santa+Isabel,+Madrid/Museo+Nacional+del+Prado,+Calle+de+Ruiz+de+Alarcón,+Madrid/Thyssen-Bornemisza+Museum,+Paseo+del+Prado,+Madrid/Plaza+de+Cibeles,+28014+Madrid/Círculo+de+Bellas+Artes,+Calle+de+Alcalá,+Madrid/Mercado+de+San+Ildefonso,+Calle+de+Fuencarral,+Madrid/Calle+de+San+Vicente+Ferrer,+Madrid/Plaza+del+Dos+de+Mayo,+28004+Madrid/@40.4173091,-3.707319,15z/data=!3m1!4b1!4m50!4m49!1m5!1m1!1s0xd422628db3a07bd:0x309814d159babf71!2m2!1d-3.6945569!2d40.4079123!1m5!1m1!1s0xd42289d66d8a2ed:0x1094f07d93ad885a!2m2!1d-3.6921271!2d40.4137818!1m5!1m1!1s0xd4228831aa2ecbf:0x5a934a7882cb528f!2m2!1d-3.6949254!2d40.4160406!1m5!1m1!1s0xd422884b69894d3:0x42597193d8bd3e47!2m2!1d-3.6932818!2d40.41951!1m5!1m1!1s0xd42288472c293df:0xa921f67f134d1c1a!2m2!1d-3.6965333!2d40.4183042!1m5!1m1!1s0xd42288866be0f79:0xc58f6e101d92b4a5!2m2!1d-3.7008111!2d40.4242361!1m5!1m1!1s0xd42286253ff1ac3:0xbe4072fd821a3125!2m2!1d-3.7049953!2d40.4258785!1m5!1m1!1s0xd422862150e1e33:0x66e5505c093f5dcd!2m2!1d-3.7041056!2d40.4267066!3e2
On the way, you see Plaza Neptune which is where Atleti celebrate their triumphs and and then Plaza Cibeles where RM fans take over when they win. It's got some of the must see views in Madrid.
For curries, cheep beer and Edge. (not my responsibility)
Head down to the Lavapíes area. Again 10 minutes from Sol. Wouldn't fuck about in this area, especially at night, but you'll get what you need.
https://www.google.com/maps/place/Lavapiés/@40.4090055,-3.7013168,19z/data=!4m5!3m4!1s0xd42262b70745fd3:0xf8234270a46d263!8m2!3d40.4084512!4d-3.7007072
Transport
I'd fuck the Metro off. You have to get a card now 5 euros then charge it with 1 or 10 trips. If you do that get one per five people and charge the 10. Then pass the card each time for each person. It'll be a mare though at the station if loads of blues are trying to work out this system plus the tourists.
I'd use the Taxis. Especially if there are more than 3. It's just not worth the difference in price. And if there are more things Madrid has than bars, it's taxis.
Stadium
As for drinking up at the stadium. I'd go east and north of it. The road at the at the South of the ground is where their Ultras meet in the bars. I' don't think the police let anyone through anyway, but you never know.