I'm going to hazard and say it's not very likely, more to human reaction time than anything. If one has to assume that the computers are not available for the quick launch sequence, one also has to accept that they are not available to control timing. Here's a blurb about the thruster capacity in the various nodes, from Spaceflight 101.
SKD, the Soyuz Main Engine, provides a thrust of 2,942 Newtons. The
entire Soyuz Attitude Control System is comprised of 28 DPO Thrusters.
Two clusters of 14 DPO Thruster are mounted on the spacecraft with 12
of these jets providing 26.5 Newtons of Thrust and the remaining 16
providing 130 Newtons.
Assuming a mass of around 7150 kg, as the same site indicates, that means that even with the smallest thruster, there's an acceleration of around 3.7mm/ second^2. Given a human reaction time of even a tenth of a second, that could result the in the delta-V being off by .3mm/s How much of a difference does that make? In 6 hours, that's only 6.5m. That doesn't sound so bad really, until you consider how long of a thrust is likely required. It's hard to know exactly, but I have found that to support a close orbit like this, a 30 degree phase angle or less is required. I don't have the exact delta v required, but I'm going to guess that it is around 10 m/s. For that length of time, one would have to do a burn with the smallest thruster around 45 minutes. I suspect the reaction time would be far worse for that length of time, but it could still theoretically be done. I'm sure the main engine is used more than the small ones, greatly reducing the thrust time required (By a factor of 100), but that would leave more uncertainty in the location.
Okay, so let's say you get your angle off by 1 degree, which would be an easy thing to do without assistance. What does that do? That would given an error of around 1.6%. That means the planned 10m/s burn could be off by as much as 16 cm/s, which over 6 hours would mean an error of around 5km, fairly significant!
Also keep in mind that the thrust requirements grow the longer the thrust takes to plan, and things get very complicated very quickly!
Bottom line, unless you have access to some kind of a computer, it seems very unlikely that you could pull this off. There might be a person or two who could pull this off, but I doubt it.