Understanding this simple circuit
The final voltage will be 1 V, yes, but you've got the time constant wrong. The time constant is Rp \$\cdot\$ C, where Rp = R1 and R2 in parallel. I know this is counterintuitive, but it's all Thévenin's fault. Look up "Thévenin equivalent".

One way to get some understanding why R2 also plays a role: suppose R2 is 1 ohm. Would the capacitor be charged more quickly? (The answer is yes!)