No (well, not exactly), because you don't know how much longer he'll be on the outbound part of the procedure turn, and you also don't know the exact spot where he started the PT. In general, the aircraft will turn outbound at or just past the final approach fix, do the 45-degree turn to the outbound leg of the procedure turn, and fly that heading for 30-60 seconds before making a 180-degree turn back to the inbound leg. Even at an airport where I have a transmitter on the ground and can thus talk to the pilot directly, that's not a very long time to read an entire clearance eg. "C-FMVY is cleared to the Toronto Buttonville airport via direct blah blah blah, maintain 5,000, depart runway 12, turn right on course, squawk xxxx, contact Toronto Centre airborne on this frequency, clearance cancelled if not airborne by (time)" and so on. Then even after I do that the pilot has to read it back, change to the UNICOM frequency to announce his intentions, taxi onto the runway and take off. It might work, barely (by which I mean there wouldn't be a collision on the runway) but there's no guarantee of any separation so it would count as a technical loss of sep every time. At most airports, I don't even have the luxury of being able to talk to the pilot directly, and am relaying the clearance through an FSS or being patched through to the pilot on a landline phone; in either case that adds more delay.
In that situation, however, if we really need to get the departure out and the arriving aircraft doesn't mind, we can use a non-radar separation standard called reciprocal track over a fix (controllers usually refer to this as "tail to tail"). Basically the arrival reports northwest of the NDB/VOR, northwestbound on the outbound leg of the approach, and I say "approach clearance cancelled, maintain 4,000, continue outbound on your present track." Then, assuming the departure is going to use a runway that will have him depart in a southeasterly direction, I can give him departure clearance and have him fly runway heading until he reaches 5,000 feet (and thus has altitude separation with the arrival). This way, the arrival is northwest of the NDB/VOR heading northwestbound, the departure is southeast of the NDB/VOR heading southeastbound, ergo they can never hit. But they have to both be flying consistent tracks, with no turns, until they get vertical. In other words I can't give the arrival a hold somewhere northwest of the NDB, because then he'd have to turn southeastbound, toward the departure. Likewise I can't have the departure take off in a northwesterly direction, even if I give him an immediate turn southeast once airborne. So if the runway is 12/30 and the departure wants to go off 30 for whatever reason, this doesn't work.
Even when it does work, you're looking at the arrival aircraft flying at least an extra 20-30 miles (probably an extra 10-15 while the departure gets his clearance, takes off, and passes through 5,000, then another 10-15 miles coming back toward the airport) which is like a 10-minute delay, so that's why it would be REALLY unusual to do this, and far easier to just wait for him to land (if he's just turning outbound then he should land within 5-6 minutes) and then roll the departure. It would only happen if the two aircraft were company and insisted on it, or if the departure was a medevac. Even in the latter case, you'd normally get a strip for the medevac departure at least 30 minutes or so before he actually called, which would have a proposed departure time on it. So you'd be able to plan this better and keep the arrival high and issue a hold (rather than an approach clearance) while the medevac departed underneath (eg. hold the arrival at 8,000 and give the medevac 7,000 until he's laterally clear).
At a towered airport, this is a lot easier -- the rule for the tower we work with is that any departure must be airborne before the arrival reaches 5 miles from touchdown. We have radar right down to the ground at this airport and tower has a drop on it, so they can time it perfectly and move traffic way more efficiently. In a situation where it's going to be really close, they'll already have given the pilot his clearance, taxi instructions, etc., and all I have to do as the radar controller is release the departure. So I can clear an arrival for the approach, hit the hotline, and tell tower "Jazz 7865 is released, runway 12, cancelled when Air Bravo 700 reports DURLU" and they'll say "ok show Jazz rolling now." They say "cleared for takeoff" and like 20 seconds later he's airborne, no messing around with full readbacks and cancellation times etc.. But you can't have radar to the ground everywhere, and it costs a lot of money to have a tower controller, so at the vast majority of the airports that I work it's some kind of one-in-one-out, non-radar separation similar to the above example.
Hope this clears it up -- non-radar separation isn't the easiest thing to understand (nor the most logical, often) so I tried to keep it relatively simple. The tail to tail over a fix standard is only one of 20 or so that we use on a daily basis, but it's one of the easiest to apply.